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Binary matroids of branch-width 3
Dharmatilake, Jack Sidathdam, Ph.D.
The Ohio State University, 1994
Copyright ©1994 by Dharmatilake, Jack Sidathdam. All rights reserved.
U-M-I 300 N. Zeeb Rd. Ann Arbor, MI 48106
BINARY MATROIDS OF BRANCH-WIDTH 3
DISSERTATION
Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University
by
Jack Sidathdam Dharmatilake, B.Sc., M.S.
*****
The Ohio State University 1994
Dissertation Committee: Approved by
G. Neil Robertson
Thomas A. Dowling
Dijen K. Ray-Chaudhuri Adviser Department of Mathematics Phifip J. Huneke Copyright by Jack Sidathdam Dharmatilake 1994 To the memory of my mother Indraiii, sister Leetheiidra, aunt Pulsara Abeywickrama and friend Ranjan Herath Guneratne ACKNOWLEDGEMENTS
First and foremost, I express my sincere appreciation to Professor G. Neil Robertson, for his mathematical insight, patience and encouragement. I also wish to thank the other members of my committee, Professors Thomas A. DowHng, Dijen K. Ray-Chaudhuri and J. Phihp Huneke for their helpfid suggestions and comments. Chapter 7 consists of joint work with Professors Neil Robertson, EUis Johnson and Sunil Chopra. I am grateful to Dr. Daniel P. Sanders for checking my computer program. Special thanks go to Ms. Terry England, for helping me with 1 ]e^ , and encouraging me to learn touch-typing. I also thank the Ohio State University and the Ohio Supercomputer Center for making their resources available to me. I acknowledge use of the software packages v^^ m VITA May 14, 1947 ...... Born - Colombo, Sri Lanka 1971 ...... B.Sc. (Mathematics) -University of Sri Lanka (Colombo) 1979 ...... M.S. (Mathematics) -Michigan State University 1985 ...... M.S. (Mathematics) -Ohio State University 1971-1975 ...... Assistant Lecturer (Mathematics) -University of Sri Lanka (Colombo) 1975-1977 ...... Teaching Assistant (Mathematics) -University of Pittsburgh 1977-1979 ...... Teaching Assistant (Mathematics) -Michigan State University 1979-Present ...... Graduate Teacliing Associate (Mathematics) - Ohio State University PRESENTATIONS 1. Branch--width of a matroid, (20 minutes), Ohio State - Denison Conference, Denison University, GranviUe Ohio, May 1992. 2 . Non-binary matroid obstructions to branch-width decompositions, (15 min utes), MAA Oliio section spring meeting, Miami University, Oxford Ohio, April 1994. FIELDS OF STUDY Major Field: Mathematics Computer Science iv TABLE OF CONTENTS DEDICATION ...... ii ACKNOWLEDGEMENTS...... iii VITA ...... iv INTRODUCTION ...... 1 CHAPTER PAGE I. Preliminaries...... 6 1 .1 . Set theory and algebra ...... 6 1.2. G rap h s ...... 6 1.3. Ternary tre e s ...... 11 1.4. M atro id s ...... 19 1.5. Lower ideals ...... 20 1.6. Representable m atroids ...... 23 1.7. Separations of graphs and matroids ...... 24 II. Branch-width of a Matroid ...... 35 2.1. Introduction...... 35 2.2. Complete-separabihty ...... 48 2.3. Branch-width of representable matroids ...... 52 III. Tangle Number and a Min-max Theorem for Matroids ...... 58 3.1. Matroid tangles ...... 58 3.2. A lemma about submodular functions...... 67 3.3. The min-max theorem ...... 70 IV. Branch-widths of a Graph ...... 78 4.1. In tro d u c tio n ...... 78 4.2. Definition of /3'{G) and elementary consequences ...... 78 4.3. Comparison of P'{G) < 2 and f3{M{G)) < 2 ...... 83 4.4. A result for 3-connected graphs ...... 8 6 V. The Lower Ideals D eterm ined by Branch-w idth < 2 ...... 94 5.1. Introduction...... 94 5.2. The lower ideals and S^q ...... 94 5.3. The lower ideals Sfi and ...... 97 5.4. The lower ideal % ...... 100 5.5. One- and two-summings of binary matroids ...... 104 5.6. A reduction to 3-connected graphs and matroids ...... 108 VI. A Structure Theory for 3-connected Matroids of Branch-width 3 ...... 109 6.1. Introduction...... 109 6 .2 . Small structures...... 109 6.3. Complete 3-separabihty of sets, of size < 5 ...... 113 6.4. Six structures...... 114 6.5. Apphcations to small binary matroids ...... 117 6 .6 . An upper bound for |5(M )|, for simple M 6 . ^ 3 ...... 119 6.7. A pphcations to 3-connected g rap h s...... 122 VII. The Lower Ideal % ...... 127 7.1. Introduction...... 127 7.2. The locahzability of triads and triangles of graphs 128 7.3. Wye-delta 3-summing ...... 136 7.4. The wye-delta 3-sum generators for % ...... 140 7.5. Robust separations...... 148 7.6. Obtaining robust separations by pushing edges ...... 151 VI 7.7. Wye-delta 3-sum decomposability of the wheel Wn, for n > 6 ...... 157 7.8. Wye-delta 3-sum decomposabihty of a general non-generator ...... 158 7.9. Obstacles to % ...... 163 VIII. The Lower Ideal ^ 3 ...... 174 8 .1 . Introduction...... 174 8.2. Five summing operations ...... 175 8.3. An important consequence of the locahzability of 4-structures...... 180 8.4. Regular matroids ...... 184 8.5. Generators of ^ 3 ...... 189 8 .6 . Obstacles of ^ 3 ...... 197 IX. Non-binary Obstructions to Branch Decompositions ...... 203 9.1. Introduction ...... 203 9.2. Uniform matroid obstacles to branch-width ...... 205 9.3. Obstacles with highest branch-width for a fixed rank .. 207 APPENDICES A. Program Output...... 215 B. An Addition Table for Coded Binary Vectors ...... 271 C. Outhne of the Computer Program ...... 279 BIBILIOCRAPHY ...... 288 vii LIST OF FIGURES FIGURES PAGE 1.1. The wheel neighborhood of n ...... 10 1.2. The first six ternary trees ...... 11 1.3. The ternary tree T with edges directed towards u ...... 14 1.4. Ternary tree with central edges ci and eg ...... 15 1.5. The branches at away from e, in T ...... 19 1.6. The lower ideal ^ ...... 23 1.7. A graft corresponding to Fy ...... 26 1.8. The separation (H,K) and the subgraph H of G ...... 27 1.9. The wheel W4 with labelled vertices ...... 33 2.1. The end-trees of an edge of a branch decomposition ...... 36 2 .2 . Constructing (T,I) from (T i,/i) ...... 41 2.3. Constructing (T,/) from {T\J\) and (TrJr)...... 43 viii 2.4. A Euclidean representation and a branch decomposition of f y ...... 46 2.5. Rank versus branch-width of where n > 2 ...... 47 2 .6 . Constructing an optimal branch decomposition of F j ...... 52 3.1. Part of a tree-labelling (T, a ) ...... 69 3.2. Sets L{u, e) and L{v, e), for incidences {u, e) and (v, e) of T ...... 71 4.1. Edges and end-trees of (T,/) ...... 80 4.2. Obtcdning (Ti, /i ) from (T,I)...... 82 4.3. Excluded minors of G, for /3'{G) < 1 ...... 84 4.4. Two separations {H\, Ho) of G ...... 8 6 4.5. Impossible separations of G ...... 8 8 4.6. Graphs G w ith \E{G)\ < 9 and P{M{G)) = P'{G) = 3 ...... 89 4.7. Graph G and graphic optimal branch decomposition ( T ,0 ...... 90 4.8. The instances n = 2 and n = 1 ...... 90 4.9. Construction of (Ti,/i) ...... 91 4.10. Graph G and graphic optimal branch decomposition (T,/)...... 92 ix 4.11. The graph G and branch decomposition (T,l)...... 93 5.1. The lower ideal ...... 96 5.2. The lower ideal ...... 96 5.3. The graph G is a 1 -sum of G% and Gv ...... 97 5.4. The lower ideal ...... 99 5.5. The lower ideal S/j ...... 99 5.6. The graph G is a 2 -sum of G% and Go...... 100 5.7. Obtaining two 2 -sum m ands G\ and Gg of G ...... 1 0 2 5.8. The lower ideal ...... 103 6.1. The graphs corresponding to the small structures ...... I l l 6.2. The graphs Gg,^ such that Mg f = M(Gg,f), for 1 < z < 4 ...... 116 6.3. The graphs JQ, and the 3-prism ...... 122 6.4. A wye-delta and a 6 -structure in G...... 125 6.5. The Graphs of the polygon matroid obstacles to branch-width 3 ...... 125 6 .6 . ATg and M(Kg) ...... 126 7.1. The lower ideal % ...... 127 X 7.2. Two significant graphs ...... 130 7.3. The branches of T at t ...... 131 7.4. The triad Y in the graph G ...... 132 7.5. Obtaining a localization of Y for a graph G ...... 132 7.6. The triad Y in the graph G ...... 133 7.7. Obtaining a localization of Y for a graph G ...... 133 7.8. The graph G with a forbidden octacube minor with Yfixed .... 134 7.9. The graph G, and obtaining a localization of y ...... 134 7.10. The graph G, and obtaining a localization of y ...... 135 7.11. The graph G has a forbidden augmented 1 ^ 3 3 minor with Y fix e d ...... 136 7.12. A wye-delta if-extension Gi at (z, {y,z})...... 137 7.13. Wye-delta H- and Jf-extension minors of G ...... 137 7.14. The graph G is a wye-delta 3-sum of G\ and G 2 ...... 138 7.15. Obtciiiiing {T,l) from (Tj,/}) and (T 2 ,Z9 ) ...... 140 7.16. A 3-sum of 2 octacubes...... 140 7.17. A big 3-separation {H,K) of the graph G ...... 141 xi 7.18. The graph W5 and a big 3-separation {H\,H2 ) ...... 142 7.19. Two pairs (Gj, Gg) of non-summands of ...... 142 7.20. A 2 -separation of G ...... 143 7.21. The unique 1 0 -edge non-generator ...... 144 7.22. The graph with vertex sequence (5,5, 3, 3, 3,3) ...... 145 7.23. Graphs with vertex sequence (5,4,4,3,3,3) ...... 145 7.24. Graphs with vertex sequence (4,4,4,4, 3,3) ...... 146 7.25. Graphs with vertex sequence (4, 3,3,3, 3,3,3) ...... 146 7.26. The octacube ...... 147 7.27. The graph G has a robust separation [H,K)...... 148 7.28. A wye-delta ff-extension Gi at (a, {6 , c } )...... 149 7.29. The wye-delta 77-extension G\...... 150 7.30. An example of a push ...... 152 7.31. The first instance for the application of ttj...... 152 7.32. The second instance for the apphcation of ttj ...... 153 7.33. A pre-push separation for 7T2 ...... 154 XU 7.34. Two post-push separations for 7T2 ...... 154 7.35. A 2-separation of G ...... 156 7.36. The big 3-separation (H,K) of G ...... 158 7.37. The graph G as a part-wheel Tf-extension ...... 159 7.38. The graph G' and its separation {H', K')...... 160 7.39. The graph G and its wye-delta iJ-extension minor ...... 160 7.40. The graph G and its subgraph H ...... 161 7.41. A separation (ff, K) of G ...... 161 7.42. The graph G and its subgraph H ...... 162 7.43. The 3-vertex cut-set G of G such that w(G — U) > 3 ...... 165 7.44. The excluded subgraphs H for a new obstacle G ...... 166 7.45. The possible wheel neighborhoods for x ...... 166 7.46. The new obstacle G and its subgraph H'...... 167 7.47. The new obstacle G with contradictory forbidden subgraph .... 167 7.48. The paths Li, Lo and L\ih\ Q ...... 168 7.49. The new obstacle G with a contradictory subdivided cube ...... 169 xni 7.50. The wheel neighborhood of x and the subgraph H ...... 170 7.51. Two minor maximal graphs G\ and G2 for a new obstacle...... 171 7.52. The minors G2a and G2b of G g ...... 172 8.1. Pictorial representation of an i\ig-sum, with relevant minors ...... 177 8.2. Two geometric representations of N4 = AG{3,2)...... 178 8.3. The matroid M, a Fano-sum and a Fano-dual-sum ...... 180 8.4. Pictorial representations of M, M[ and ...... 181 8.5. Obtaining (T,/) from {TiJ\) and {T2 J 2 ) ...... 182 8 .6 . The branch decomposition (f/,, 183 8.7. The matroid R \2 as a graft ...... 186 8 .8 . The graph Ffg 3 with edges labeled by vectors ...... 190 8.9. The matroids (M(/\ 3 ^3 ))* and (M (/i 3 ^ 3 + e))* as grafts ...... 191 8.10. The matroids Fj and Fj as grafts ...... 192 8.11. The matroids # 3 and N\q as grafts ...... 193 8.12. The matroids and A^isg as grafts ...... 194 8.13. Sixteen 3-connected generators of ^ 3 , as graphs or grafts ...... 195 XIV 8.14. The graphs of and Vg with edges labeled by vectors ...... 198 8.15. The matroids {M{K^))* and (M(Vg))* as grafts ...... 199 8.16. The matroids iîjo &nd jVgg as grafts ...... 200 8.17. The matroids N n and as grafts ...... 201 8.18. Ten obstacles to 33^ as graphs or grafts ...... 202 9.1. Some uniform matroid obstacles to branch decomposition ...... 207 9.2. A geometric representation of the matroid N ...... 213 A.I. The graphs corresponding to the non-generators in List A .l ...... 217 A.2. A histogram for probabihties of base sizes ...... 268 A.3. A histogram for probabihties of column pseudo-orbit sizes ...... 270 C.l. The generation of 3-connected binary matroids by the program ...... 283 XV LIST OF LISTS LIST PAGE A.I. Polygon matroids of 3-connected grapliic/ cographic summands. .223 A.2. Non-regular 3-connected matroids with at most 12 elements (that are in or are obstacles to ^ 3 ) ...... 225 A.3. Obstacles to branch-width 3 (three) ...... 259 A.4. Self dual matroids (38) ...... 259 A.5. Matroids that have a 6 -structure (20)...... 262 A.6 . Matroids that have a non-localizable 3-structure (37) ...... 263 A .7. Active generators ( 6 ) ...... 264 A.8 . Matroids of decomposition-type A 3 (five)...... 265 A.9. Matroid of decomposition-type A 4 ...... 265 A.10. Matroids of decomposition-type Fy (sixteen) ...... 265 XVI LIST OF TABLES TABLE PAGE A.I. Branch-width distribution ...... 259 A.2. 4-structure distribution (for matroids of branch-width 3) ...... 261 A.3. 3- and 6 -structure distribution (for matroids of branch-width 3 ) ...... 261 A.4. Non-localizable 3-structure distribution (for matroids of branch-width 3 ) ...... 262 A.5. Generator distribution...... 264 A.6 . Decomposition-type distribution ...... 265 A.7. Statistical information about the distribution of the number of bases...... 267 A.8 . Statistical information about the distribution of the number of column pseudo-orbits...... 269 B.l. The addition table ...... 271 xv ii INTRODUCTION In this thesis we study two matroid invariants called branch-width and tangle number, both of which are defined in terms of separations. In our discussions greater emphasis is placed on branch-width than on tangle number. Both these concepts were first defined and developed for graphs by Robertson and Seymour in [4]. This introduction offers a summary of this thesis, highhghting significant results. Although the hon’s share of this thesis is about binary matroids there are some results about non-binary matroids also. In Chapter 1 we develop some results that wül be useful in the discussions of branch-width and tangle number. Ternary trees are introduced because branch- width is defined in terms of these trees. A ternary tree is a tree in which all non-pendant vertices have valency 3. The pendant vertices are referred to as leaves. The central-edge-lemma and the httle twig lemma are two very useful results about ternary trees. The central-edge-lemma states that every ternary tree T contains a central edge c such that each of the two end-trees of c contains at least one third of the leaves of T as vertices. An important consequence of the Httle twig lemma states that if a ternary tree T has at least 5 leaves then T has an edge e such that one of the end-trees of e has exactly 4, 5 or 6 leaves of T as vertices. Several results about matroid separations also appear in this chapter. For the rest of this introduction let M and G denote a matroid and a graph, respectively. Also, let S{M) and p denote the ground set of M and the rank function of M, respectively, and let M[G) denote the polygon matroid of G. We define branch-width for matroids in Chapter 2. In a branch decomposition, we label the leaves of a ternary tree with the elements of a matroid M. The width ru of a branch decomposition of M is defined to be the maximum order of 1 2 a separation in the family of separations of M that is specified by the end-trees of the edges of the associated ternary tree. The branch-width of M, which is denoted by /?(M), is the minimum of the widths of branch decompositions of M. Branch-width is preserved under duahzation and is monotone with respect to taking minors. The branch-width of M is bounded above by 2 if and only if M is the polygon matroid of a series-parallel graph. While P{M) < p{M) 4 - 1 in general, if M is binary and p{M) > 2 then < p{M). The concept of comi)lotc separabihty, which is a reformulation of branch-width, is also introduced in this chapter. Chapter 3 is the only chapter in wliich we discuss tangle number. The tangle num ber of M, which is denoted by t{M), is the maximum order of a tangle of M. A tangle of order t of M, is an assignment to every separation of M of order less th an t, of a “small side” and a “big side”, satisfying certain axioms. The following two main results are proved in tliis chapter. If M has no coloop and P{M) ^ 1 then P{M) — t[M). This is known as the min-max theorem and cstabhshes the equivalence between branch-width and tangle number in most cases. If M is /c-connectcd then P{M) = k if and only if M has precisely k — 1 distinct tangles. In Chapters 4 to 8 we discuss only binary matroids and graphs. For the rest of this introduction, let M be a binary matroid. In Chapter 4 we introduce 2 kinds of branch-width for a graph G. One of these is P[M{G)). The other, which is denoted by P'{G), is defined by using the vertices of G to determine the orders of the separations of G. It is proved that if G is 3-connected then P{M{G)) = 3 if and only if P'{G) = 3. Using this result we may sometimes argue in terms of vertices, in order to prove resnlts about graphs of branch-width 3. A lower ideal of graphs or matroids is a class of graphs or matroids, respectively, that is closed under isomorphism and taking minors. We are interested in the lower ideals determined hy p < k or P' < k, where k is a non-negative integer. In Chapter 5, we discuss the 5 lower ide;ds determined by < 0 , < 0 , /3 < 1 , P' < I and P < 2. Each of these lower ideals is generated by a finite set of generators using a finite set of proper closed summing operations. Also, each of these lower ideals has a finite set of obstacles. A summing operation is proper if each of the two summands of a sum is a proper minor of the sum in a certain way. A summing operation is closed on a lower ideal if every sum defined by that summing operation, of any two elements of that lower ideal, is also in that lower ideal. In Chapter 6 we introduce the theory of small structures for 3-connected binary matroids. Small structures are subsets ofS{M) which are 3-separated from their complements, and are characterized by their matroid structure. A ^-structure has exactly k elements, for k = 3,4,6. The small structures consist of 2 kinds of 3-structures known as triangles and triads, 2 kinds of 4-structures known as wye- deltas and 4-circuit/bonds and 2 kinds of 6 -structures. A triangle is a 3-circuit, while a triad is a 3-cocircuit. Both kinds of 4-structures consist of 4 elements. A wye-delta contains a triangle as well as a triad, while a 4-circuit/bond is a 4-circuit as well as a 4-cocircuit. A 6 -structure consists of6 elements. If A is a 6 -structure in M then either the restriction of M to A is isomorphic to the polygon matroid of K2,3 ^.nd the restriction of M* to A consists of two disjoint triangles, or vice-versa. The following results are proved using the theory of small structures. K M is 3-connected and A is a subset of M which is 3-separated from its complement with |A| < 6 then A is completely 3-separable. If M has at most 9 elements then /3{M) < 3 . If M is 3-connected and M consists of 10 elements then M is an obstacle to branch-width 3 if and only if M has no 4-structure. If M is 3-connected and has branch-width 3 then the number of elements of M is bounded by a hnear function of the minimum of the rank and the corank of M. In Chapter 7 we develop a structure theory for the lower ideal of graphs of branch-width < 3. AU such graphs can be produced from 15 generators, using a set of 4 proper closed summing operations. The generators are the 7 non- separable graphs of at most 3 edges, the wheels of 3, 4 and 5 spokes, the 3-prism and its dual, A ' 3 3 and the graph obtained by joining 2 non-adjacent vertices of 7 ^ 3 3 , and the octacube. The summing operation that is used to generate 3- connected graphs of branch-width 3 is called the wye-delta 3-sum. There are 4 graphs that are obstacles to branch-width 3. These are the cube, the octahedron, A’5 and Vg- order to prove that wye-delta 3-summing is closed it suffices to prove that every wye-delta of a graph of branch-width 3 is localizable. This proof is accomplished by showing that in a graph of branch-width 3 there is exactly one minor-minimal instance in which a triangle is not-localizable and exactly two minor-minimal instances in which a triad is not localizable. The concept of a big 3-separation plays a vital role in this chapter. A big 3-separation of a graph G is simply a 3-separation {A,B) such that|A| > 5 < \B\. The basic lemma for decomposabihty is that if G is not isomorphic to W 5 and G has a big 3-separation then G is wye-delta 3-sum decomposable. In this sense ld ' 5 is an exceptional generator. In order to prove that a non-generator G is decomposable we must show that G has a big 3-separation which satisfies certain other properties. Such a big 3-separation is called a robust separation of G. If G is a non-generator and a given big 3-separation (iï, K) of G is not already a robust separation then we can always obtain a robust separation from (JT, K) by pushing a single edge from one side of this separation to the other. Finally, in order to prove that there are only 4 pairwise non-isomorpliic graphs that are obstacles to branch- width 3, we assume that there is a “new” obstacle. By using Wagner’s structure theorem for excluding a Ag-minor we show that this new obstacle must be planar. Thereafter, the result about wye-delta 3-sum decomposabihty in the presence of a big 3-separation becomes useful, once again, since an obstacle cannot have a big 3-separation. In Chapter 8 we consider the lower ideal determined by binary matroids of branch-width < 3. For general binary matroids, that are not necessarily regular, there are 5 summing operations. Three of these are generahzations of wye-delta 3-summing, which is one of the graphic summing operations. These summing operations involve the 3 distinct binary matroids on 8 elements. In addition to these there are 2 other summing operations, which involve the Fano matroid and its dual, respectively. A computer program, running over all non-regular binary matroids that consist of at most 1 2 elements, is used to obtain the hkely set of generators and obstacles. Two conjectures are made which, if true, would estabhsh the given set of 22 generators as being complete. There are 10 obstacles to branch-width 3. There is some evidence that this set of obstacles is also complete, and this conjecture is made here. In Chapter 9 we find the complete fist of uniform matroids that are obstruc tions to branch decompositions. Tliis chapter also contains a finiteness result for another class of obstacles. Appendix A consists of a version of the program output, while Appendix C gives an outhne of the nature of the computer program. Besides these, an ad dition table for coded binary vectors is also provided in Appendix B. This table may be used for hand-checking the program output. C H A P T E R I Preliminaries 1.1. Set theory and algebra Throughout we adopt the standard set theoretic conventions; set membership, intersection, union, disjoint union, difference, symm.etric difference, inclusion and proper inclusion are denoted by G, ft, U, Ù, \, A, Ç and respectively. The empty set is denoted by 0 . We shall consider finite sets almost exclusively. The cardinality of a set Y is denoted by |F|. When Y Ç X, we may write instead of X\Y. Also, we may simply write x instead of {%}, when no confusion may be caused by doing so. T he power set of X is denoted by 2 ^ . The sets of integers, positive integers and real numbers are denoted by Z, and E, respectively. A function f, from X to y, is denoted by / : A — > Y. If g is a prime power, we denote the Galois field that consists of q elements by GF(g). 1.2. Graphs In what follows the elementary graph theoretical notation of [ 8 ] is adopted. We indicate that two graphs G\ and G2 are isomorphic by writing G\ = Gq- If G is a graph, we denote the vertex-set and edge-set of G by V{G) and E[G), respectively. A graph without loops or multiple edges is called a simple graph. T he complement G^ of a simple graph G is the simple graph with vertex set V{G), two vertices being adjacent in G'^ if and only if they are not adjacent in G. (This differs from the notation of [ 8 ]). Suppose V 6 V{G)^ U Ç F(G), e 6 E{G) and A Ç E{G). We denote the valaicij of V in G by valg(n) or val(i;), if G is understood. The set of vertices of G that are adjacent to v is called the neighbor set of v and is denoted by Nq{v) or N{v). The edgeless subgraph of G with vertex-set U is denoted by [[/]. The subgraphs of G induced by U and A are denoted by G[U] and G ■ A, respectively. We denote G[F(G)\17] and (G)\{u}] by G — U and G' —u, respectively. The set of vertices of attachment of a subgraph if of G is denoted by W{G,H) or by W[H), if G is understood. So W[G,H) = V{H n (G ■ (E{G)\E{H))). The subgraph F of G defined by V{H) = {V{G)\V{H)) U W{G,H) and E{H) = E{G)\E{H), is called the relative complement of H in G. The vertex n of G is said to be 3-joined to 3 other vertices a, b and c of G if there exist 3 internally vertex-disjoint paths from v to a, b and c, in G. The subgraph F of G is said to be 3-joined to 3 vertices a,b,c ^ V{H) if 3 distinct a',b',c' G V{H) with disjoint paths from a', b' and o' to a, b and c, respectively. Paths here have no “repeated” vertices. The set of all (connected) components of G is denoted by G(G) and we let w(G) = |G(G)|. G is said to be edge-transitive if its automorphism group acts transitively on E{G). If G is edge-transitive then the graph obtained from G by deleting an arbitrary edge of G is denoted by G\e. If G is simple and G^ is edge-transitive then the graph obtained from G by joining two arbitrary non-adjacent vertices of G is denoted by G -f- e. In order to facilitate the definition of a minor we introduce some notation. /(j(e) = {w G V{G) : w is incident with e}. = {d G F(G) : /G(d) Ç [/}. Definition. Let F be a subgraph of G and S Ç {V{H))\E{H). A minor M of G is a graph that satisfies the following conditions: (1) y(M) Ç G(F). (2) E[M) = S. (3) If cZ G 5 and fcid) — {x,y} then fMid) = {W, F } , where A'’, F G V{M) with X G V{X) and y G V{Y). Informally speaking, the minors of G are formed from G by deleting isolated vertices, deleting edges and contracting edges. It is worth noting that the order in which edge-deletion and edge-contraction are carried out is immaterial to the resulting minor, up to a certain type of isomorpliism. The minor of G obtained by deleting all the edges of A is denoted by and the minor of G obtained by contracting all the edges of A is denoted by G'^. If = {e} then we simply denote these minors by and G”, respectively. We define 5p(G), the spanning tree graph of G, by letting its vertices be the spanning trees of G and defining two such spanning trees to be adjacent if and only if one can be obtained from the other by deleting one edge and adding another. T he empty graph (or null graph)^ the circuit on n vertices, the complete graph on n vertices, the wheel on n + 1 vertices and the complete bipartite graph w ith maximal independent sets of vertices having m and n vertices, respectively, are denoted by fl. G», A'n, Wn and Km,ni respectively. We define bridges of a subgraph as follows. Definition. Suppose H is a subgraph of G. The subgraph of G that is induced by an edge which is not in E{H) but whose incident vertices are in V{H), is called an inner bridge of H. Suppose V{H) 7^ V{G), and let G be a component of G — V{H). Let F be the set of all edges of G, each of which is incident with a vertex of C as well as with a vertex of H. Then the subgraph C U (G ■ F) of G is called an outer bridge of H. If % is a subset of a topological space the closure of X is denoted by cl(X). We discuss plane embeddings. Definition. Suppose G is a (topologically) closed subset of and V is a finite subset of U. If the components of U\V are homeomorphic to open fine segments and are finite in number then T — {U,V) is said to be a plane embedding. The sets of vertices, edges and faces of T are denoted by V^(T), E{T) and T’(T), respectively, with V(T) = V, E{T) being the set of all components of U\V, and F{T) being the set of all components of K^\i7. The graph G (T) of T is defined to have vertex-set V and edge-set £J(T), with vertex v incident with edge A if and only if u G cl(A). If G is planar connected graph then we denote the dual of G by (3*. We state Euler’s Theorem (without proof), and prove an elementary but useful consequence. Lemma 1.2.1 (Euler’s Theorem). Suppose T is a plane embedding such that G(T) is nonempty and connected. Then (1-2) |V(T)|-|E(T)|-f |F(T)| = 2. a Definition. Suppose G is planar. Then G is said to be cosimple if G* is simple. Proposition 1 . 2 . 1 . Suppose G is a nonempty connected planar graph that is simple as well as cosimple. Then either G or G* has a trivalent vertex. Proof. Assume otherwise. Then every vertex of G as well as G* has valency at least 4. Let T be a plane embedding such that G = G'(T). Then 4|E(T)| < 2|E(T)| > 4|F(T)|. Now from Euler’s Theorem it follows that 0 > 2, which is a contradiction. B Before defining wheel neighborhoods, we define the relations of “incidence” and “adjacency” as they pertain to plane embeddings. Definition. Suppose T — ([/, V) is a plane embedding, and a G F (T ) U E{T) U E(T). We define t{a), the type of a, by t{a) — V li a E E (T ), t[a) = E \i a e ET) and t{a) = F if a G F(T). Incidence is defined between elements that are not of the same type. Suppose a, 6 G y ( T ) U E{T) U F(T). Then a and h are said to be incident if and only if t{a) ^ t{b) and, a Ç cl(6 ) or 6 Ç cl(a). Suppose a 6 , and a and b are not incident. Suppose also that A G {E, E , T’} with t[a) ^ A ^ t( 6 ). T hen a and b are said to be A-adjacent if 3c such that t{c) = A and c is incident with both a and b. 10 We note that the relation of incidence between a and b is defined only when a and b are in distinct sets of F(T), E(T) and F(T). Definition. Suppose T — (17, V) is a plane embedding, x e V and G = G (T ). The subgraph of G formed by the vertices and edges of G that are incident or adjacent with x is called the neighborhood graph of x. If a neighborhood graph is isomorphic to a wheel, with the rim possibly subdivided, then it is called a wheel neighborhood. (See Figure 1.1.) F ig u re 1 . 1 . The wheel neighborhood of v We state a fundamental result about plane embeddings (without proof). L e rn m a 1 . 2 .2 . Suppose T = {U,V) is a plane embedding. Then (1) If G{T) is connected and non-null then every component of R^\U except the unbounded region is homeomorphic to an open disk, and the unbounded region itself is homeomorphic to a planar annulus. (2 ) If G{T) is non-separable and has at least 1 circuit then all face boundaries of T are circuits. (3) If G{1l) is 3-connected and has at least 6 edges then every vertex of G (T ) has a wheel neighborhood, with hub of valency at least 3, and rim possibly subdivided. We suggest BoUobas [1], Bondy and Murty [2] and Tutte [ 8 ], as general refer ences for graphs. 11 1.3. Ternary trees Definition. A vertex u of a tree T is called a leaf if val(r>) = 1 . We denote the set of leaves of T by L[T). We say that T is a ternary tree if the valency of every non-leaf vertex of T is equal to 3. The bridges of the single-vertex subgraph r[u], of T are called the branches of T at v. In our illustrations of graphs we depict vertices by small solid circles except for leaves of trees which may be represented by open circles. If a vertex or an edge is denoted by a certain symbol then that symbol is written next to that vertex or edge, respectively. If a leaf of a tree has a label then that label is written inside the open circle wliich represents that leaf. The subtle distinction between a symbol that represents a leaf and the label of a leaf wiU become clearer in Chapter 2. The first six ternary trees are given by Fig 1.2. Figure 1.2. The first six ternary trees We prove some results about ternary trees. Our first result states 2 identities. Proposition 1.3.1. If T is a ternary tree then |F(T)| = 2|L(T)| 2 and |E(T)| = 2|T(T)|-3. Proo/. 2|E(T)| = E«6 F(nvai(u) = |T(T)| -f 3(|y(T)| - |T(T)|) = 3|y(T)| - 2|T(T)|. Since T is a tree, |P(T)| = \V{T)\ - 1. Therefore, 2|F(T)| - 2 = 3|y(T)| - 2|T(T)|. Hence \V{T)\ = 2\L{T)\- 2 and \E{T)\ = 2|T (T )| - 3. H To show that the converse of Proposition 1.3.1 is false we shall construct a non-ternary tree T with e edges and A leaves, where e = 2A — 3, with A > 3 as follows. Let T[ be the star that has A leaves. Then |P(Ti)| = |L(Ti)| = A. Because A > 3 it follows that T\ is not a ternary tree and e > A. Now form a 12 tree T from Ti, by subdividing the edges of T\ until there are e edges. Then T is as required. The following definition is useful when we discuss subgraphs of a ternary tree. Definition. Suppose T is a ternary tree and H is a subgraph of T. T hen the set of all vertices of H that are also leaves of T is called the set of T-leaves of H or simply the T-leaves of H. We prove that the leaves of a ternary tree may be distributed as we please. Proposition 1.3.2. Suppose pi, pg and pg are positive integers. Then 3 a ternary tree T and a trivalent vertex v of T such that the 3 branches of T at v have exactly p\, po and pg vertices, respectively, which are also leaves of T . Proof. We prove this by induction on p = pi + pg + pg. If p = 3 then the result is true. Assume that it is true when p — t, where t is a positive integer > 3 and suppose p = t +1. Since t +1 > 4 we may, without loss of generality, assume that Pi > 2. Then p% — 1, pg and pg are positive integers and (pi — 1) + P 2 + P3 — Therefore, by the induction hypothesis, 3 a ternary tree Ti and a trivalent vertex V, of Ti, such that the 3 branches Y\, Yg and Yg, of Ti at n, have exactly pi — 1, pg and pg vertices, respectively, which are also leaves of Tj. Now spht any Ti-leaf Y%, into 2 leaves to obtain a ternary tree T as required. ■ From Proposition 1.3.1 it follows that all ternary trees have an even number of vertices. If we consider the ternary spanning trees of 7ig„ where n is a positive integer, we obtain the following result. Proposition 1.3.3. If n is a positive integer then the set of all ternary spanning trees of Kon » an independent set of vei'tices of 5p(/fg,j). Proof. If 71 = 1 then the result is trivially true. Suppose n > 2. Let Tg,T 6 V{Sp{K2n)): where Tg is ternary and T is adjacent to Tg. It suffices to prove th a t T is not ternary. T can be obtained from Tg by deleting an edge e of Tg and adding an edge / which was not in Tg. Let the vertices adjacent to e be u and v. Since T Tg 13 the edge / is not adjacent to u or it is not adjacent to v. Therefore, either v alj’(u) 7^ 3 or val'p(r) ^ 3. Hence T is not a ternary tree. ■ For 1 < n < 2 , the set of all ternary spanning trees of K2n forms a vertex cover of Sp{K2n)^ but for n > 3 this set does not form such a vertex cover. This is because for n > 3, there is no ternary spanning tree of K 2n which is adjacent in Sp{K2n)-, to any spanning path of iv 2n- The next proposition wiU, in turn, enable us to prove the centrcil-edge-lemma, which is a useful result. Proposition 1.3.4. Suppose T is a tree that has at least one edge and let the edges o fT be directed (arbitrarily). Then G V{T) such that every edge incident with u is directed towards u. Proof. Assume that such a vertex u does not exist. Then we may construct a function / : V{T) — > E{T), by defining f{w) to be any edge incident with w that is directed away from w, for every vertex w of T. Then / is injective because a directed edge cannot be directed away from two distinct vertices. Therefore, |y(T)| < |£^(T)| which is a contradiction because |F(T)| — |F^(T)| + 1. B Definition. Suppose T is a tree and e G E{T). Then a component of T( is called an end-tree of e (in T). Proposition 1.3.5 (Central-edge-lemma). If T is a ternary tree then 3e G E{T) such that each of the two end-trees of e has at least one third of the leaves of T, as vertices. Proof. If \E{T)\ = 1 then the result is trivially true. Suppose |£^(T)| > 1. Then T has more than 2 leaves. If e G E{T), let u%(e) and V2{e) be the two vertices that are incident with e. Let the end-tree of e that contains u,(e) as a vertex, be denoted by T{{e), for i = 1,2. The number of T-leaves of Tj(e) is equal to |F(Tj(e)) fl T(T)|, for i = 1,2. If |F(Ti(e)) n L{T)\ < |y(T g(e)) fl L{T)\ then direct e from iq(e) to W 2 (e). In case of a tie choose a direction arbitrarily. Then, by Proposition 1.3.4, there is a vertex u of T such that every edge that is incident with u is directed towards 14 u. Since T has more than 2 leaves the unique edge that is incident with any leaf wiU be directed away from that leaf. Therefore u is not a leaf. Let c\{u),C2{u) and C3 ( 7t) be the 3 edges of T that are incident with u. W ith out loss of generality, let the end-tree of C{{u) that contains w as a vertex be T2 (cf(u)), for 1 < i < 3. (See Figure 1.3). Figure 1.3. The ternary tree T with edges directed towards ii Since Ti{c\{u)),Ti{c2{u)) and T%(c 3 (u)) are pairwise disjoint, and every leaf of T has to be a vertex of one of them. (1-3) E|F(Ti(cXn)))nL(T)| = |L(T)|. 2=1 Therefore, without loss of generality, |y(ri(c,(u)))nL(r)| > Since ci(u) 3 is directed towards ?/, i ^ ( n i (1-4) |y(T 2 (ci(«)))nL(T)| > |y(ri(c,(«)))nL(T)| > Therefore, the edge ci(u) satisfies the conclusion of the central-edge-lemma. H Definition. An edge of a ternary tree T that satisfies the conclusion of the central-edge-lemma is called a central edge of T. We denote the set of all central edges of T by Z{T). 15 Figure 1.4 illu.strates a ternary tree that has 2 central edges. Cl Figure 1.4. Ternary tree with central edges c-i and eg Proposition 1.3.6. If T is a ternary tree then T[Z{T)] is a subtree ofT. Proof. Let a, w 6 V{T[Z{T)]) and let P{u,w) be the unique path in T from u to w. It suffices to prove that E{P{u,w)) Ç Z[T). Let m = |F(P(w, w))|. We proceed by induction on m. If m = 0 then there is nothing to prove. Assume that E[P{u,w)) Ç Z(fP)., if m < A; — 1, where k is a positive integer. Suppose m = k. Let the sequence of vertices of P{u.,w), beginning at u and ending at w, be denoted by a = Dq, Uj,..., = w. Let the edge of P(a, w) th at is adjacent with vi_\ and a, be denoted by e,, where \ < i < k. Let d and / be central edges that are adjacent with a and a;, respectively. If £? = Cj then a, S V{T[Z{T)]) and the path in T from a^ to w consists of the k — 1 edges 6 2 ,..., e^. By the induction hypothesis, 6 3 ,..., e^. G Z(T). Therefore, E{F{u, w)) Ç Z{T). If / = then a similar argument prevails. Suppose d 7^ Cj and / 7^ e^. Then 3a_,, a^^, G y(T)\{ag, a,,..., a^^} such that a_j and a^.^, are incident with the edges d and /, respectively. If e G E{P{u,w)) U {d,/}, and e is incident with the vertices af_i and a, then let the end-tree of e which contains vj as a vertex be denoted by 2 ^ (e), for i — 1 < j < i, where 0 < f < A: -f-1 . Because there is a path from a^ to a_, that does not have e, as an edge, y(To(e, )) 3 y(T_i(d)). Similarly, because there is a path from a^ to a^.^j that does not have e, as an edge, y(Ti(eJ) D h^(dfc+i(/))- Therefore, |y(To(ej)) fl L(r)| > |y(T_](d) n L(T)| > and |y(Ti(eJ) n L(T)| > |y(Tt+i(/)) n 16 \L(T)\ Z/(T)| > —-— . Therefore, Cj G Z{T). Now, by the argument given in the case d = ei, the result follows. H Definition. T[Z{T)] is called the central subtree of T. We use the central-edge-lemma to prove 2 results about the existence of end-trees, with certain properties, in ternary trees. In these results, the word “twig”, refers to such an end-tree. Proposition 1.3.7 (Little Twig Lemma). Suppose T is a ternary tree with at least n-\-l leaves, where n > 2. Then 3e G E{T) with an end-tree Tc such that n < \V{Te)nL{T)\ < 2n-2. Proof. Let A(T) = |L(T)1. If n 4- 1 < A(T) < 2n — 1, we simply let e be any pendant edge. To discuss the 2 remaining cases, corresponding to 2n < A(T) < 4n — 3 and 4n — 2 < A(T), suppose c G Z{T). Let Aj and A 2 be the number of T-leaves of the 2 respective end-trees of c (in T), with Aj < Ag. Then: (1.5) s Also A% 4- A2 = A(T) and Aj < A 2 < 2Ap First, suppose 2n < A(T) < 4n — 3. Then, by (1-5) it follows that n < A2 and Ai < 2n — 2. If Aj < n — 1 and 2n — 1 < A2 then 2Ai < 2n — 2 < A 2 , which is a contradiction. Hence, either n < Ai < 2n — 2 or n < A 2 < 2n — 2, as required. Next, suppose 4n — 2 < A(T). Then, by (1-5) it follows that 2n — 1 < A 2 and n < Ai. If Aj < 2n — 2 then the result is true. So assume A% > 2 n — 2. We prove a useful claim. Claim. Let T be a ternary tree and d G E{T) such that an end-tree Tq, of d in T, contains at least 2n — 1 leaves of T as vertices. Then 3e G E{T q) with an end-tree Tg in T such that n < \V{Te) fl L{T)\ < 2n — 2. Proof. Let m = |L(To) fl L{T)\. Then, 2n — 1 < m. We prove this by induction on rn. 17 Let V be the vertex of Tq, incident with d. Let d\ and d^ be the other 2 edges of T that are incident with v. T hen, ^ 1 , ^ 2 G E{Tq). Let T\ and T 2 be the respective end-trees of d\ and c?2 in T, that do not contain n as a vertex. Also, let m j = \V{Ti) n L(T)|, for 1 < î < 2. Then, 0 < mj < m, for 1 < z < 2 and m \ m 2 = m. If m = 2n — 1 then n < m{ < 2n — 2, for some i such that 1 < i < 2, and then d{ satisfies the requirements of the claim. Now assume that the claim is true when 2n — 1 < m < p, where 2n — 1 < p. Suppose m — p + 1. Then 2n < rn. Without loss of generality, let n < m \ < p. If 2n — 1 < m \ < p then, by the induction hypothesis, 3e E E{T) such that the claim is satisfied. If n < m\ < 2n — 2 then d\ E E{T) satisfies the claim. Replacing the edge d in the claim by the central edge c completes the proof. S Proposition 1.3.8 (Disjoint Twigs Lemma). Suppose T is a ternanj tree which has at least 3n — 2 leaves, where n > 2. Then: (a) 3 e i , 6 2 E E{T), with ei and 62 having end-trees Tj and T 2 in T, respectively, such thatT\ is disjoint from T 2 and n < |R(rf)nT(T)| < 2n — 2 , for i = 1,2; (b) 3e E E{T) with e having endtrees T\ and To such that |F (T 2 ) fl ^T)|>n<|y(Ti)nL(T)|< 2 n - 2 . Proof. At the outset we observe that (a) imphes (6), by letting e = ej. So it suffices to prove (a). Because rz > 2, it follows that |L(T)| = 3n — 2 > n -f 1. Therefore, by the httle twig lemma, 3ei E E{T) with C] having an end-tree T\ in T, such that n < \V{Ti) fl L{T)\ <2n — 2. Let T2 be the other end-tree of e\ in T. Then |y(7^) fl L(T)| = |L(T)| - |y(Ti) fl L{T)\. If |L(T)| = 3n — 2 then n < |R(T 2 ) fl L(T)| < 2n —2 also. Hence, we may take 6 2 = ej to satisfy (a). So suppose |L(T)| > 3n-2. T hen |y (T 2 )nL(T)| = |L(T)|-|y(Ti)nL(T)| > 3n — 1 — (2n — 2) = n -f 1. If \V{T2) fl L[T)\ < 2n — 2 then we may once again take 6 2 = e\ to satisfy (a). 18 Lastly, suppose \L{T)\ > 3n — 2 and \V(T2) fl L{T)\ > 2n — 2. We prove a useful claim. C la im . Suppose. |L (T )| > 3n—2 and |y (T 2 )nL(T)| > 2n—2. Then 3 c2 £ E{T2), with 69 having an end-tree in T, such that e\ ^ ^(T g) and n < |l/(Tg) fl L (T )| < 2n-2. Proof. Let x = \V{T2) H L{T)\. We proceed by induction on x. Because n > 2, it follows th at a; > 3. Let v be the vertex of Tg that is incident with ej. Because To has at least 3 vertices, v is trivalent. Let eg and eg be the edges of Tg that are incident with v and let f/g and Tg be the end-trees of eg and eg, respectively, that are also subtrees of Tg. Let xi — \V{Ui) n T(T)|, for i = 2,3. Then 0 < z, < z, for i = 2,3, and 3:2 + zg = X. If z 2n — 1 then n < X( < 2n — 2, for some i such that 2 < i < 3. Hence, e, and Ui satisfy the claim. Now, assume that the claim is true, when 2n — 1 < z < p, where p > 2n — 1. Suppose z = p 4 - 1. T hen x > 2n. We may, without loss of generality, let n < X2 < p. If zg < 2n — 2 then zg and Tg satisfy the claim. If 2n — 1 < zg < p then, by the induction hypothesis, 3d £ T(Tg) with d having an end-tree Tg in T, such that eg ^ T(Tg) and n < |T(Tg) (1 L{T)\ < 2n — 2. Since eg ^ T(7g) it follows that e\ ^ T(Tg). Therefore, d and Tg satisfy the claim. The claim directly implies (a). ■ We make some additional definitions and conventions for later use. Definition. Suppose T is a ternary tree, e € E{T) and u, n are the vertices of T that are incident with e. Suppose also that v is not a leaf and /, p are the other 2 edges of T that are incident with v. Then the 2 branches of T at v th at contain / and g respectively as edges, are called the branches at v away from e. The union of the two branches at v away from e is called the (maximal) subtree at V away from e. 19 In our figures we shall draw the edges / and g of the branches at v away from e, and sometimes represent the rest of these branches by triangles. Suppose the sets of T-leaves of the branches that have / and g as edges are labelled by X and F, respectively. Then we indicate that fact by writing the label {X or Y) inside the respective triangle. (See Figure 1.5). E X = E{H) a n d /o r Y = E{K), where H and K are graphs then we may simply write H a n d /o r K inside the respective triangle. Figure 1.5. The branches at v away from e, in T 1.4. Matroids We indicate that two matroids M\ and M 2 are isomorphic by writing M\ = M 2 . The direct sum of M% and Mo is denoted by Mj © M 2 . Let M be a matroid. Unless otherwise specified, we use the following notation for M . [M] = Matroid isomorphism class to which M belongs; S{M) = Underlying set of M (finite); p = Rank function of M ; ( 1 - 6 ) ,i^(M) = Set of all independent sets of M; Sd[M) = Set of all bases of M; "^(M) = Set of all circuits of M; (M) = Set of all hyper planes of M. 20 We w rite p{M) instead of p{S{M)). We refer to a circuit or a cocircuit of M, that consists of k elements, as a k —circuit or a k —cocircuit, respectively. A 3-circuit or a 3-cocircuit may also be called a triangle or a cotriangle, respectively. Let U Ç M. Then the minor of M obtained by deleting all the elements of is denoted by M ■ 17 and the minor of M obtained by contracting all the elements of is denoted by M x 17. If 17^ = {z} then we may denote M ■ U and M X 17 by M'j. and M'J., respectively. Also, we let M\U = M ■ [U^) and M/U — M x (17^). The dual of M is denoted by M*. The rank function of M* is denoted by p*. The symbol M may be deleted when doing so causes no confusion. Also, if X e S(M) then we may simply write x instead of {%}. Our notation for some well-known matroids is given below. M{G) = The circuit matroid of a graph G; F-r = The Fano matroid; (1-7) Uk,n = The uniform matroid of rank k on a set of n elements; 17o Q — The matroid whose underlying set is 0. Let G be a graph and M = M{G). If X is a subgraph of G, then we may also write p{X) for the rank of E{X). We recommend Oxley [3], Welsh [9] and White [10], as references. We follow the defintions of [3] and [9] as closely as possible in what follows. 1.5. Lower ideals In this section we discuss lower ideals of graphs and matroids with respect to the partial order defined by the minor relation. Definition. Suppose .Sf is a class of graphs or matroids. If VX G ^ it is true that every minor of X also belongs to .Sf and every graph or matroid, respectively, that is isomorpliic to X also belongs to J 2f, then if is said to be a lower ideal (of graphs or matroids, respectively). 21 Definition. Suppose .if is a lower ideal and X ^ J5f. If every proper minor of X belongs to jSf then X is said to be an obstacle to _if. As an example, let us consider the class ^ of all forests. Then is a lower ideal with a unique obstacle, up to isomorphism. That obstacle is the graph consisting of a single loop. When discussing lower ideals we shall henceforth omit the phrase “up to iso morphism” although that phrase may be necessary for our statements to be strictly accurate. We view the structure determined by a lower ideal _Sf as consisting of two parts — the internal structure and the external structure. The lower ideal has a particularly interesting internal structure if it is generated by a finite “basis” using a finite set of “proper closed summing operations”. In that case J5f is said to be “finitely generated”. The external structure of jSf is completely determined by its obstacles. We reach the definition of finite generation via a sequence. Definition. Suppose .if is a lower ideal. Then a is said to be a summing oper ation on i f if V.4, B E Sf such that A and B satisfy the pre-conditions specified by (7 , the operation a assigns a set (r{A, B). Every element of the set a{A,B) is called a a-sum of A and B. Such an element is also said to be a-decomposable. If C E cr{A,B) then A and B are called a-summands of C. If S is a set of summing operations on i f and cr G S then a cr-sum, tr-decomposable element or (7-summand is also called a S-sum, decomposable element or H-summand, respectively. The summing operation a is proper if whenever a{A,B) is defined both A and B are proper minors of every element of a{A,B). The summing operation a is closed (on if ) if whenever A, B G i f and cr(A,B) is defined every element of (r(A, B) is in if . Henceforth, when <7 (A, B) is spoken of, it shall be assumed that A and B have satisfied the pre-conditions of a and th a t a{A,B) is hence defined. Definition. Suppose i f is a lower ideal and S is a finite set of proper closed summing operations on if. Also, suppose X G if. 22 T hen, if X is not S-decomposable then X is said to be a generator (with respect to S) . A generator X is said to be an active generator if A' is a S-summand. A generator that is not active is said to be a inactive generator. D efinition. Suppose .if is a lower ideal, S is a finite set of proper closed summing operations on .if and is a finite set of generators (with respect to S). Suppose also that A G .if. We inductively define A to be finitely generated (by ^ via S) if either A 6 or A is a 2-sum of A\ and Ag, where Aj, Ag E .if and A% and Ag are finitely generated (by ^ via S). The above definition is inductive because A\ and Ag are proper minors of A. D efinition. Suppose .if is a lower ideal, S is a finite set of proper closed summing operations on jSf and is a finite set of generators (with respect to 2). Then jSf is said to be finitely generated by the basis SB (via 2) if every element of .if is finitely generated (by SB via 2 ). As an example, we consider the lower ideal ^ defined earher. Let the summing operations in 2 be the 0-siim and the 1-sum that are defined as follows. Suppose G{ and Gg are disjoint non-null graphs. The 0 -sum of G\ and Gg consists of the graph Gi U Gg. A 1-sum of Gi and Gg is obtained from G\ U Gg by identifying any pair of vertices {ui,ug}, where v\ and vg are non-isolated vertices of G\ and Gg, respectively. If ^ consists of the nuU graph, the single vertex graph and the fink graph then it can be shown that ^ may be finitely generated by the basis SB via the two summing operations in 2. The nuU graph is the only inactive generator. In our illustrations we represent a lower ideal by a cone. The elements of the basis are drawn inside the cone closer to the bottom and the names of the summing operations are written inside the cone closer to the top. The obstacles are drawn outside the cone. Figure 1.6 illustrates the lower ideal 23 0 - and 1-sums Figure 1.6. The lower ideal ^ 1.6. Representable matroids Suppose p is a prime number and M is a matroid that is representable over GF[p). Then there are three ways in which we will present M. The first is by stating that M is representable over GF[p) and then writing down a full row rank matrix Rep(M) which is a matrix representation of M. We call this the matrix presentation of M. As an example, let us consider the matroid Ty and the matroid F ^ th at is obtained from Ty by declaring one of the circuits of Ty to be an independent set. Following Oxley [3], we call F ^ the non-Fano matroid. Then Fy is a binary matroid and / I 0 0 0 1 1 1\ (1-8) Rep(Fy) =0101011. 0 0 1 1 1 0 1 Fy is a ternary matroid and /I 0 0 0 1 1 1\ (1-9) Rep(FT) =0101011. \0 0 1 1 1 0 1/ The second method of presenting M involves the coding of the column vectors of Rep(M) as non-negative integers. If (ag, O],... , a.nŸ' is a column vector of Rep(M), then we assign the codeword 71 ( 1- 10) ^ a,p' = ug + a\p ■ -f a^p^ -f • • • -f Urip”, to that column. i=0 2 4 In order to present M, we first state that it is representable over F. This is followed by the name or number assigned to M if such exists, followed by a colon. Next, we write the ordered pair (p(M), |S'(M)|). Lastly, we write down the codewords corresponding to the respective columns of Rep(M), in bold type. We call this the vectorial presentation of M. We illustrate using F-j and F ^ . Fj is a binary matroid and (1-11) Fy : (3,7) 1 2 4 6 5 3 7 fy is a ternary matroid and (1-12) Ff : (3,7) 1 3 9 12 10 4 13 The information given here may be enhanced, if necessary. It is easy to see that one presentation may be obtained uniquely from the other. Once the characteristic of a prime field and the length (number of coordinates) of a vector over that field are specified, a non-negative integer will uniquely determine that vector via the coding process described earher. Due to this, we will be able to denote, refer to and perform vector operations on the elements of a representable matroid, by using the codewords corresponding to its elements. As an example, consider the binary matroid Fj. Instead of writing (1-13) we may write 6 -H 5 + 3 = 0, which is more compact. The third method of presenting a matroid M is a combination of the first two. Here, we once again state that M is representable over GF(p). Then we write down the matrix Rep(M), which is constructed from Rep(M) by adding a top row that consists of the codewords assigned to the respective columns, in bold type. We illustrate as before. Fj is a binary matroid and /I 2 4 6 5 3 7 \ (1-14) Rep(Fy) = 1 0 0 0 1 1 1 0 1 0 1 0 1 1 Vo 0 1 1 1 0 1/ 25 Fy is a ternary m atroid and /I 3 9 12 10 4 (1-15) E 5 ( F f ) = J Î 0 1 0 i Vo 0 1 1 10 If M is any matroid then the matrices Rep(M) and Rep(M) are not uniquely determined, but if one of these matrices is given then the other may be obtained from it uniquely, in a natural way. Therefore, when we speak of one of these ma trices, in a context of the other, we shall assume that the two matrices correspond with each other in that natural way. If y is a vector space over some field and K Ç V then we let (A”) denote the subspace of V that is spanned by K. Also if u is a vector given as an n-tuple over some field, where n G Z_|_, then the set of aU non-zero coordinates of of u is called the support of u and is denoted by Supp(u). We need the following definition, for binary matroids. Definition. A cycle of a binary matroid M is a subset of S{M) that is express ible as a disjoint union of circuits of M. While a circuit of a binary matroid is always non-empty, a cycle may be empty because there may be no circuits in the disjoint union. A binary matroid that is almost the polygon matroid of a graph, may be geometrically interpreted using a concept from Seymour [5]. Definition. Suppose G is a graph and Ç y(G). Then the ordered pair (G, A) is called a graft. With every graft {G,U) we can associate a binary matroid B{G,U) on the set E{G) U {X}, where AT is a new element. To define B{G,U) we associate with each e G E{G)L}{X} a vector v{e) over GF{2) and then specify a subset G of E{G) U {X} to be a cycle of R(G, U) if and only if SegC — 0 The vectors v{e) are defined as follows. Let y be a |y (G)|-dimensional vector space over GF(2), and {h^ : v G V{G)} be a basis of V. Then for e G E{G) U {X}, the following must hold. (1) v{e) = 0 if e G E{G) and e is a loop. (2) v{e) = bu + bx, ii e E E{G) and e is incident with u and v, where u ^ v. (3) r,(X) = E,eyf'"- 26 We illustrate a graft (G, U) by drawing the graph G and drawing small circles around the vertices of U. Figure 1.7 illustrates a graft for Figure 1.7. A graft corresponding to F-j We follow Oxley [3], in defining the concept of an alfine matroid. Definition. Suppose X = {a;i,..., is a multiset of elements of the vector space {GF{q)Y, where g is a prim e power and r is a positive integer. Then X is said to be affinely dependent if n > 1 and ...,«» E GF{q), not all zero, such th a t a{Xi = 0 (in {GF{q)Y) and Y Ji= \ = 0 (in GF{q)). The multiset X is affinely independent if it is not affinely dependent. Clearly, if X is affinely independent then X must be a set. Suppose 5 is a set that indexes a multiset of elements of the vector space {GF{q)Y. Let ./ be the collection of subsets / of 5 such that I indexes an affinely independent subset of {GF{q)Y■ Then it can be proved that is a collection of independent sets for a matroid on S. This matroid is called the affine matroid on 5. We denote the affine matroid on {GF{q)Y by AG{r,q). 1.7. Separations of graphs and matroids In this section we derive some results about separations that will be useful later. We shall subsequently define the concepts of a separation of a matroid and the order of such a separation. We shall obtain these definitions by generalizing the analogous graphic concepts. 27 Definitions. Suppose G is a graph. If H and K are edge-disjoint subgraphs of G such that H \J K = G then [H^K) is said to be a separation of G. The order of the separation (H,K) in G is denoted by {G, H, K), and is defined to be the number of vertices of G that H and K have in common. Therefore, (1-16) ('(G,.^,Ai = |y(ff)ny(AH. When there is no risk of ambiguity, we write instead of ('{G,H,K). In our illustrations we usually depict a part of a graph or the whole graph by a bounded region that is enclosed by a simple closed curve. Suppose 77 is a subgraph of a graph G and H is represented by the region R that has the simple closed curve G as its boundary. Then the symbol H may be written outside R near the curve G, or inside R. The vertices of attachment of H are indicated by dots drawn on G. We indicate that H has at least n edges by drawing n hnes inside R. To show that 2 vertices a and b are non-adjacent we draw a fine from a to 6 and cut that fine. If we wish to indicate that a vertex has valency at least 2 in 77 we write a plus sign ( 4-) near that vertex. In Figure 1.8, the graph G has a separation (77,77) of order 3, with V{H fl k) = w}. Both 77 and K have at least 3 edges each. The vertices v and w are not adjacent to each other in 77 and the valency of u in 77 is at least 2. + H w G H Figure 1.8. The separation {H,K) and the subgraph 77 of G 28 Matroids have no vertices. Therefore, if vve are to generalize, we must cast the definition of the order of a separation in terms of the rank, rather than in terms of the number of vertices. The next result is a step in this direction. Proposition 1.7.1. Let G be a graph and M = M{G), the polygon matroid of G. If {H, K ) is a separation of G such that neither H nor K is a null graph then (1-17) Zf, A:) = + X ^ ) - XG) + w( -h w(K) - w(G). Proof f'{G,Hjq = \v{H)nv{K)\ = |y(ff)|+|y(K)|-|y(ffuÆ)| = \V{K)\- |y(G)| = p{H) + uj{H) + p{K) + lj{K) - p{G) - uj{G). a If we assume that G, H and K are all connected then oj[H) = cj[K) = w(G) = 1. Therefore, by equation (1-17) it follows that ^'{G^H,K) = p{H) + p{K) — p{G) + l. We are now ready to proceed with the definitions for a general matroid M. Definition. If A and B are complimentary subsets of S{M) then {A,B) is said to be a separation of M . The order of the separation (A, B) in M, is denoted by ^{M, A, B) and is defined by ^(M ,A,B) =XA)+XB)-XM)-bl, if Aÿ60^ B = 0, otherwise. We note that if A ^ 0 ^ B then (,{M,A,B) > 1 by the submodular inequality for the rank of a matroid. {A,B) is called a k-separation of M, where k is a positive integer, if |A| > k < |B | and f{M, A, B) < k. A /.’-separation (A, B ) is exact if A, B) = k. The matroid M is k-separable if it has a ^-separation. M is k-connected if it has no /c'-geparation for any k' such that 1 < k ' < k. If (A, B) is any separation of M then A^ B) = ^{M,B,A). When there is no risk of ambiguity, we shall write B) instead of ^{M, A, B) and we m ay also write A, B) or f*{A, B) instead of ^{M*^A, B). If A Ç 5{M) then A determines the separation (A, A*^) of M. We adopt the convention B = A^. Also, if A; C 5(M), for some subscript i then let B{ = Af. 29 For a singleton subset {z} of M we adopt the notation Y — S{M)\x. Also, if X{ E S{M), for some subscript i then let = 5'(M)\zj. Suppose G is a graph with no isolated vertices and H is an edge-induced sub graph of G. Then we see that H is the edge-induced subgraph G- (E{G)\E{H)). Proposition 1.7.2. Let G he a connected graph and (H,K) be a separation of G. Then [E{H), E{K)) is a separation of M{G) and (1-19) 4(M(G), E(;ï), )) < ('(G, jiT). Proof. T h at {E{H), E{K)) is a separation of M{G) follows directly. Also, if either H or K is a null graph then the result is true as <^(M(G), E{H), E{K)) = 0 = f'{G, H, K). So we may assume that neither H nor K is nuU. Because G is a connected graph, w(G) — 1. Also uj{H) > 1 < ea{K). Therefore, from Proposition 1.7.1 it follows that ^'(G, H, K) > p{H) p{K) — p{G) -b 1 -f-1 — 1 = p(E(^)) +X ^(K )) -p(M(G)) -b 1 = ((M(G),B(^),f;(Fr)). N In equation (1-19), strict inequahty may occur if one of the subgraphs is non- nuU but edgeless or if it is disconnected. As an example, consider G = K^, H = [F(A'5)] and K = /V5. T hen ff{G,H,K) = 5 b u t f{M[G),E{H),E{K)) = 0. As a further example, let G = A' 5, H = G ■ {e,, and K = G ■ (A (/i 5) \{ e ,, 62}), where e% and eg are not incident with a common vertex. Then f'{G,H,K) = 4 but ^(M(G), E{H), E{K)) = 2-b4-4-bl=3. If G is a graph and {H,K) is a separation of G, we may write ^(G, If, Ji) or f{H,K) instead of ^{M{G), E{H),E{K)). We prove some results about matroid separations. Proposition 1.7.3. If {A,B) is a separation of M then it is also a separation o/M * ond r (/I, ^ ) = ((;!, B). Proof. Suppose {A,B) is a separation of M. Because S{M*) = S{M) it follows th at {A, B) is a. separation of M* also. EA = 0 o rB = 0 then ^*(A, B) = 0 = 30 £,{A,B). Otherwise, r + /(g) - /(M) + 1 = |A| - (f(M ) - / g ) ) + |g| - (f(M ) - /,(A)) (1-20) -(|g(M)|-/M)) + l = P (^ ) + p{B) - p{M) + 1 = ^(A,g). B Proposition 1,7.4. Suppose M is a matroid with at least 2 elements, x G S{M) and A C 5 (M ). Then (1) If X is a loop or a coloop then = 1, otherwise f[x,Y) = 2. (2) f[A,B) = 1 if and only if A is the union of (connected) components of M and 0 ^ ^ S[M). Proof. (1) If z is a loop then p{x) = 0 and p{Y) = p{M). Hence, ^{x,Y) = 0 + p{M) — p{M) + 1 = 1. If z is a coloop then the result follows from Proposition 1.7.3 because z is a loop of M*. If z is neither a loop nor a coloop then p{x) = 1 and p{Y) = p{M). Hence, ((z, P ) = 1 + p{M) — p{M) + 1=2. (2) First, suppose A is a union of components of M and % ^ A ^ S{M). Then since p{A) + p{B) = p{M) it follows that f{A,B) = 1. Next, suppose f{A,B) = 1. Then 0 ^ A S{M) and p{A) + p{B) = p{M). Hence, A is a union of components of M (see Theorem 5.2.4 on page 71 of [9]). fl Proposition 1.7.5. If{A,B) is a separation of M then (1-21) ((M,A,g) < l+m in{p(A),p(g),/(A),/(g)} < l + min{|A|,|g|}. Proof. The result follows from the inequality |J\T| > p{X) < p{M) VX Ç S(M) and Proposition 1.7.3. B Proposition 1.7.6. Suppose M ' is a minor of M and (A, g ) is a separation of M . If A' = A n S{M') and B' = B C\ S(M') then (A% g') is a separation of M ', ond ^(M% A', g') < ^(M, A, g ). Proof. Suppose (A, g ) is a separation of M. Because A n g = 0, it follows that A' n g' = 0 also. Hence, (A% g') is a separation of M '. 31 By induction, it suffices to prove the result when |5(M )\5(M ')| = 1. By passing to the dual and using Proposition 1.7.3, which imphes that ^*(M, A, B) = ^(M , A, B) and ^*{M' ,A ', B') = A' ,B'), we may, without loss of generality, assume that M' = M^, for some x G S{M). If = 0 or = 0 then the inequahty is trivially true. So suppose A! ^ 0 7^ B'. Without loss of generahty, assume that x E A. Then A' = A \{z} and B' = B. We consider the two cases when z is a coloop of M and it is not, separately. We denote the rank function of M' by p'. First, suppose z is a coloop of M. Then p'{M) = p{M) — 1 and p'(A') = p{A) — l. Therefore, ^{M',A',B') = p{A)-l+p{B)—p{M) + l-i-l = ^(M, A,B). Next, suppose x is not a coloop of M. Then p'(M) = p{M) and p'{A') < p{A). Therefore, ^(M', A', B') < + p(B) - p(M) + 1 = ((M, A, B). = Proposition 1.7.7. M has a non-loop circuit if and only if there exists a sepa ration (A,B) of M such that f{A,B) > 1. Proof. Suppose first that M has no non-loop circuit. Then by Proposition 1.7.4, it follows that if (X,Y) is separation of M then ^{X,Y) < 1. Conversely, suppose M has a non-loop circuit C. Let z E C and A = {z}. Then z is neither a loop nor a coloop. Therefore, by (1) of Proposition 1.7.4, it follows that ^(j4, B) = 2. ■ Proposition 1.7.8. Sxippose {A,B) is a separation of M such that \A\ > 1 and |B | > 2. Also, suppose h E B. Then (1-22) |^ (A ,B )-((A U 6 ,B \6 )|< 1 . Proof. Because |A| > 1 and |B| > 2, it follows that f{A, B) = p{A) + p{B) - p{M) 4 -1 and i{A U 6, B\h) = p{A U h) 4- p{B\h) - p{M) 4- 1. Hence, (1-23) B) - U 6, B\6) = p(A) - p(A U 6) -b p(B) - p(B\&). Since —1 < p{A) — p{A U 6) < 0 and 0 < p{B) — p{B\h) < 1, the result follows. B 32 Proposition 1.7.9. Suppose M is a k-connected matroid, where k is an inte- ger>2. Lei m = min{ [5(M )/2J, fc — 1}. Then (i) M* is also k-connectcd; (ii) M has no circuit or cocircuit that has < m elements; (Hi) If A Q S{M) such that |A| < m then A is independent in M as well as in M *, and A^ spans M as well as M* ; (iv) Neither M nor M* has a separation of order 1; (v) Suppose (A, B) is a separation of AI such that |A| < \B\, and t is an integer such that 2 < t < m. Then ^{M,A,B) — t if and only if ]A| = t — 1. Proof, (i) This follows from the fc-connectivity of M and Proposition 1.7.3. (ii) By (i), it suffices to prove the result for circuits. Assume 3A £ 9f(M) such th a t |A| = t < m. Then |B| = |A^| > m > t. Therefore, by the ^-connectivity of M it follows that ^(M, A, B) > f + 1. Hence, p{A) + p{B) — p(M ) + 1 > f -(- 1. Therefore, t — 1 +p{B) — p{AI) > t and from this it follows that p{B) > p{AI) + \, which is a contradiction. (iii) Since the assertion is true if A = 0, we assume that A is nonempty. That A is independent in M as well as M*, follows from (ii). It suffices to prove that B = A^ spans M. Since |A| < m, it follows that |B| > m > |A|. This implies th a t p{B) = p{M), because otherwise ^(M, A, B) = p{A)-\-p{B)-p{M) + l < |A| and this violates the ^-connectivity of M. (iv) This follows from Proposition 1.7.3. (v) First suppose i{AI,A,B) = t. Then |A| < t — 1, because otherwise (A,B) would be a f-separation of M, violating the fc-connectivity of M. By Proposition 1.7.5, it follows that |A| > f — 1. Therefore, |A| = f — 1. Conversely, suppose |A| = t — 1. Then, by (iii), p{A) = |A| = t — 1 and p(B) = p{M). Therefore, ^{M,A,B) = t. B Suppose M is a binary matroid and (A,B) is a separation of M. Then it is possible to obtain a matrix representation of M that is useful in the context of (A,B). Such a representation is obtained by appropriate column clearing. 33 Definition. Suppose P is a binary {mxn) matrix and Q is an (mxni) submatrix of P, where n\ < n. Suppose also that the (row) rank of Q is r. Then Q is said to be upward column cleared (respectively, downward column cleared) if the ((m — r )x 7i) submatrix of Q consisting of the first (respectively, last) m — r rows of Q, is a zero matrix. Definition. Suppose M is a binary matroid and [A,B) is a separation of M . Suppose also that P is a fuU row rank matrix that is a matrix representation of M. Then P is said to be a matrix representation of M with respect to the separation (A,B) if the first \A\ columns of P correspond to the elements of A (hence the last |P| columns of P correspond to the elements of P), the submatrix of P that consists of the first |A| columns of P is downward column cleared and the submatrix of P that consists of the last \B\ columns of P is upward column cleared. Such a matrix P may be denoted by Rep(M, A, B). Also, if a top row of codewords that correspond to the respective column vectors is added to a matrix R ep(M , A, B) then that matrix may be denoted by Rep(Af, A,B). We note that neither Rep(M, A,P) nor Rep(M, A,P) is unique. As an example, consider a binary matroid M = W4, where the vertices of W4 are as labelled in Figure 1.9. Let A consist of the 3 edges that are incident with the vertex a, and the edge that is incident with the vertices c and h. Let B consist of the other 4 edges of W4. c a d Figure 1.9. The wheel W4 with labelled vertices Then we may write / I 1 1 0 0 0 0 O' 0 10 110 10 (1-24) Rep(M,A,P) = 0 0 1 0 1 0 0 0 \0 0 0 0 0 1 1 1. 34 Proposition 1.7.10. Suppose M is a ^-connected binary matroid and {A,B) is a separation of M . Then in any m,atrix R ep(M , A, J5) the number of rows in which there is a nonzero entry in one of the first |A| coordinates, and there is a nonzero entry in one of the last |R | coordinates, is equal to — 1. Proof. Let n be the number of rows of Ke-p{M,A,B) that satisfy the speci fied property. Also let a and b denote the number of non-zero rows of the two submatrices of Rep(M, A, P) that correspond to A and B, respectively. Then n = a + b — p{M) = ^(A, B) — 1. ■ CHAPTER II Branch-width of a Matroid 2.1. Introduction Robertson and Seymour, in their monumental series of papers entitled “Graph Minors”, introduced and developed several invariants for graphs, based on the separations of these graphs. In [4], they defined branch-width and tangle number for graphs. In that paper they also introduced branch-width for matroids. In this section we repeat their definition of branch-width, derive some consequences and do some examples. All the “width”-concepts developed by Robertson and Seymour look at sepa rations in the context of “tree-labeUings”. In the case of branch-width, the trees used are ternary and the tree-labeUing is a bijection which assigns the elements of a matroid (or the edges of a graph) to the leaves of such a tree. The two sets of labels, on the two respective end-trees of any edge of a labelled tree, are complementary. Therefore, they can serve as a separation. Thus every edge of a labelled tree induces a separation of the matroid (or the graph). To any edge of a labelled tree we assign the order of the separation that is induced by that edge. To the tree-labelhng, we assign the maximum of the values assigned to the edges of the labelled tree. Finally, the branch-width itself is taken to be the minimum of the values assigned to all tree-labeUings of that matroid (or graph). We now define formally. Definitions. Let M be a matroid. If M has less than 2 elements then the branch-width of M is defined to be zero. Otherwise, we proceed as follows: 35 36 An ordered pair of the form (T,/), where T is a ternary tree with |Z,(T)| = |5(M)| and I : L(T) — > S{M) is a bijection, is called a branch decomposition of M. We shall call I a leaf-labelling of T by M. Also, if t; is a leaf of T then l{v) is called the label of v (in {T,l)). Suppose e S E{T). Then we let Ae{T,l) and Be{T,l) denote the sets of labels of the leaves of T, wliich are vertices of the two respective end-trees of e in T. When there is no risk of ambiguity, we may denote these two sets by Ae and Bg, respectively. We refer to the end-trees of e, whose sets of T-leaves are labelled by Ac and as the A-end-tree of e and the 5-eird-tree of e, respectively. See Figure 2.1. .4-end-tree of e Z?-end-lree of e Figure 2.1. The end-trees of an edge of a branch decomposition We now define the order of the edge e, the width of the branch decomposition (T, I) and the branch-width of the matroid M, which are denoted by tf{M, T, /, e), ^ ( M , r , 1 ), and ^{M), respectively. 0 (M , T ,/, e) = f{M,Ae,Be) = p(Ae) + p{Be) - p{M) -f 1 ; \k(M , T,I) = max{V'(M, TJ,e) ■. e E E(T)}; (2-1) P{M) = m ini's(M ,T,/) : (T,/) is a branch decomposition of M} = min{max{^(M, T,l,e) : e E E{T)} : (T, I) is a branch decomposition of M}. If e £ E{T) where (T, /) is a branch decomposition of M then we may refer to e as an edge of this branch decomposition. If (T, /) is a branch decomposition of M such that $(M , T, Z) = /3{M) then it is called an optimal branch decomposition 37 (of M). When there is no risk of ambiguity, we may denote and '^{M,T, l) by i/)(e) and ^ (T , Z), respectively. From the definition it is clear th at /3(f/o,o) = — 0 because each of these matroids has fewer than 2 elements. Now suppose M is a 2-element matroid. Let S{M) = { a , 6 }. If (T, 1) is a branch decomposition of M then T consists of a single edge e and the two leaves of T are labelled by a and 6 , respectively. Therefore, (2-2) = p{a) + p{h) - p{M) + I. From this it follows that every 2 element matroid except bas branch-width 1 , and 0{Uio) = 2 . Proving that P{M) = n directly, where n is a positive integer, involves two steps. The first is to prove that P{M) < n. This may be accomplished by constructing a branch decomposition of M, whose width is equal to n. The second step is to prove that P{M) > n. For the larger matroids this second step usually involves the central-edge-lemma. Our first result is helpful when testing a branch decomposition for a particular upper bound. Proposition 2.1.1. Suppose (T,l) is a branch decomposition of a matroid M and n G Then the following are equivalent. (1) $(T,Z) (2) ip{e) < n Ve 6 E{T) such that min{|Ae|, \Be\} > n. (3) 4y{e) < u Ve e E{T) such that imn{p{Ae),p*{Ae),p{Be),p*{Be)} > n. Proof. The result follows from Proposition 1.7.5. B In the above, we have stated (2) because it is easier to find cardinality than rank. We next consider a matroid which is the union of 3 sets, aU of whose ranks are bounded. We show that the branch-width of such a matroid may also be bounded, appropriately. Proposition 2.1.2. Let M he a matroid. (1) If 3Fi,F2,F^ Ç S{M) such that S{M) = Fj U F 2 U T 3 then (2-3) P{M) < 1 + max{min{p(Ff),p*(F;)} : 1 < f < 3}; 38 (2) Suppose p{M) > 2 . Then j3{M) < p{M) ifandonlyif 3F\,F'2-,F^ Ç. S{M) such that each F{ is contained in a hyperplane for 1 < z < 3, and S{M) = F\ U i^2 U -F3 . Proof. ( 1 ) We may, without loss of generality, assume that the F{ are nonempty and pairwise disjoint. By Proposition 1.3.2, it follows that 3 a ternary tree T w ith a trivaient vertex v so that we may label the sets of T-leaves of the 3 branches at V, by the elements of Fp F2 and F 3 , respectively. Let this leaf-labelling be denoted by /. If e G E{T) then either Ae{T, I) or Be{T, I) has rank or corank less than or equal to m =ma.x{mm{p{Fj), p*(Fj)} : 1 < i < 3}. Then, by Proj^osition 2.1.1, it follows that ^'(T,/) < 1 + rn. Hence the result follows. (2) First, suppose that 3F;,Fg,F 3 Ç S{M) such that each F, is contained in a hyperplane for 1 < i < 3, and S{M) — F\ U F 2 UF 3 . Then, by (1), it follows that /3(M) < 1 + max{p(Fj) : 1 < i < 3} < 1 4 - p(M) — 1 = p{M). Conversely, suppose (3{M) < p[M). Let (T,/) be an optimal branch decom position of M. For every e G F(T), we may without loss of generality assume th a t p{Ac) < p{Be). Now we direct each edge e towards its end-tree whose set of T-leaves is labelled by Be- Then, by Proposition 1.3.4, it follows that 3z G V{T) such that all the edges at z are directed towards z. If z is not trivalent then p{M) = 2 and S{M) may be expressed as required. If z is trivalent then let Fj, Fg and F 3 be the 3 subsets of S{M) that label the sets of T-leaves of the 3 respective branches at z. We claim that p{F{) < p{M) — 1 for 1 < i < 3. For, suppose, without loss of generality, that p{F\) = p{M). T hen by the choice of z it follows that p(F) U F 3 ) = p{M) also. Then ’L(T, Z) > p{M) -f p{M) — p{M) -t-1 = p(M) + 1 , which is a contradiction. Hence the result follows. Remark. To see that the pre-condition p{M) > 2 is necessary in (2), let M consist of a loop and a coloop. Then /?(M) = 1 — p{M) but S{M) cannot be expressed as required. B The next result examines the behavior of branch-width in rega.rd to familiar matroid constructs. 39 Proposition 2.1.3. Let M and N be matroids. Then (1) IfN^M then /3{N) = f3{M); (2) (3) // |g(M)| > 2 (Aen 1 < /3(M) < 1 + min{H:9(M)|/3l,XM),XM*)},- (4) If M\ is a minor of M then P{M\) < P{M); (5) Suppose |5(M)| > 3 and x E S{M). If M\ = M ’^ or M\ — M'f then (6 ) If Ml is obtained from M by deleting some of the loops and some of the coloops of M and |5(Mi)| > 2 then (3{M\) = (3{M); (7) If M\ is an underlying simple matroid of M and j3{M\) > 2 then f3{M\) = (8 ) Suppose S{N) = S{M). Suppose also that N has a rank function a such that cr{X) < p{X) y X Ç S{M), and a{S{M)) = p{S{M)). Then (5{N) < (9) If C \t .. are the connected components of M and 3i with 1 < i < k such that \Ci\ > 2 then P{M) — max{^(M • Cf) : 1 < z < k}. Proof. ( 1 ) Because the branch-width of a matroid is defined in terms of its rank function which is isomorphism invariant, the result follows. (2) If |5(M)| < 1 then /3{M*) = 0 — l3{M). Now, suppose |5(M)| > 1. Let (T,/) be an optimal branch decomposition of M. Then (T,/) is also a branch decomposition of M*. Then, by Proposition 1.7.3, it foUows that if e G E{T) then (2-4) r, Z, e) = Ae,Bg) = ^(M, Ag, Bg) = V(M, T,Z, a). Hence, $'(M*,T, Z) = Ÿ(M, T, Z) and < (i{M). Replacing M by M* yields /3(M) < f3{M*). Hence the result follows. (3) Since |5'(M )| > 2 it foUows th at 1 < (i{M). Suppose (T, Z) is an optimal branch decomposition of M and e G E{T). Then, by Proposition 1.7.5, it follows that ^(e) < 1-t-min{p(Ae),p*(Ag)}. Therefore,/3(M) < 1-|-min{p(M),p(M*)}. Since M can be expressed as the union of 3 subsets, each of which has at most [|5(M )|/3] elements, by (1) of Proposition 2.1.2, it follows that jS{M) < 1 -|- n ^ w i / 3 1 - 40 (4) If W Ç S{M) then M* x W = {M ■ W)*. Hence, by (2), it suffices to prove the result when Mi is of the form M ■ W. Also, by induction, it is enough to consider the case M% = where x G S{M). If |5(M)| < 2 then 0{Mi) = 0 < /3{M). Suppose |5(M)| > 2 . Let (T,l) be an optimal branch decompositionof M. Obtain the ternary tree Ti from T by deleting the leaf u of T where l{v) — x, along with the edge that is incident with v and suppressing the resulting divalent vertex. Define the bijection /i : L[T\) — > S{M\) by letting /| = / f L{T\). Then (Ti,/i) is a branch decomposition of M\. Suppose Cj e E[T\). Let us denote Ae^{Ti,l\) and He,(T’i,/i) by A\ and B\, respectively. Then 3e G E{T) such that Ai = Ae{TJ) fl S'(M i) and B\ — Be{TJ) n S{Mi). Then, by Proposition 1.7.6, it follows that (2-5) = ((Ml, Ai,Bi) < ((M,Ae(T,Z),Be(T,/)) = rA(M,T,/,e). Hence, ;9(Mi) < $(M i,Ti,/i) < »(M,T,Z) = /)(M). (5) By (4), it suffices to prove that P{M) - 1 < Also, by (2), it suffices to prove the result when Mi = M^, because M'J — Let pj be the rank function of M\. Since |5(M )| > 3, we have that |5'(Mi)| > 2. Let (Tp/i) be an optimal branch decomposition of Mi and let i> G L{T\) with incident pendant edge c. Obtain a ternary tree T from T\, by attaching two pendant edges b and d to v. Let the leaves of T corresponding to b and d be u and respectively. Define a bijection / ; L{T) — y S{M) by letting l{z)=l\{z), ii z e L{T\)\v ( 2 - 6 ) = h{v), if z = u = z , if z = It). (See Figure 2.2) Let e G E{Ti). Then e G E{T) also. Let us denote Ae(T, /), Be{T, /), Ac{Ti, l[) and j5e(Ti,/]) by A, B, Ai and B\, respectively. Without loss of generality, let A = Ai U {z} and B = B\. T hen V’(M ,T ,/ , e) = p(A) + p{B) — p{M) -f 1 = p{A\ U { z} ) 4 - p{B\) — p{S{M{) U { z } ) 4 - 1. If z is not a coloop of M then V;(M,T,f,e) < p,(Ai) -b 1 4- p,(^l) - 4-1 = V (M i,ri,/i,e) 4-1. If z 41 (T,l) w Figure 2.2. Constructing (T,l) from (Ti, /%) is a coloop of M then ^(M , T, I, e) = p,(Ai ) + 1 + Pi(-Si) - Pj (5(M j)) — 1 + 1 = Hence, the result follows. Remark. To see that the condition |S'(M)| > 3 cannot be dispensed with, let M = 2 _ Then j3[M) = 2 b ut j3[Mi) = 0. (6 ) By (4), it suffices to prove that j3{M\) > (3{M). By (2), we may assume that only loops are deleted from M, in order to form M\. Also, by induction, we may assume that only one such loop q is deleted. Let be an optimal branch decomposition of Mj and let v E L{T\) have incident pendant edge c.Obtain a ternary tree T from T\ by attaching two pendant edges b and d to v. Let the leaves of T corresponding to b and d he u and w, respectively. Define a bijection I : L{T) — )■ S{M) by letting f(z) = f](z), ifz 6 L (T i)\u (2-7) — li{v), if z = u — q , if z — lu . Then rp{M,TJ,b) = •0(Mi, T j,/i, c) and ■ip{M,T,l,d) = 1. Also, if e 6 E{T) fl E{Ti) then = ■^(Mi,Ti,/i,e). Because |5(Mi)| > 2 it follows that > 1. Therefore, /3(Mi) = ^(M i,Ti,/i) = $(M,T,Z) > /3(M). 42 Remark. To see that the condition |S(Mi)| > 2 is necessary we consider a ma troid M that has 2 elements, at least one of which is a loop. If M\ is obtained from M by deleting a loop then /3{Mi) = 0 although f3{M) = 1. (7) Once again by (4), we have that P{M\) < j3{M). So it suffices to prove that (3{M\) > (3{M). By (6 ), we may assume that M is loopless. By induction, it suffices to prove the result when Mi is obtained from M by deleting one element p of a pair } of parallel elements of M. Let (Ti,Zi) be an optimal branch decomposition of Mj and v G L[T\) such that l\{v) = Pj. Let c be the pendant edge that is incident with v. Obtain a ternary tree T from T\ by attaching two pendant edges b and d to v. Let the leaves of T corresponding to b and d be u and w, respectively. We now proceed as in the proof of ( 6 ), defining a bijection I : L{T) — > S{M) by letting Z(z) = Zl(z), i f z G L ( r i ) \ n (2-8) ~ Pi 1 if z = n = p , if z — w . Then ^ (M , T, /, 6 ) = ^(M , T, /, d) = 2, and if e G E{T) 0 E[Ti ) then ik{M,T,l,e) = 0 (M i,Ti,/i,e). Therefore, max{/?(Mi),2} = 'S{M\,Ti,li) > P{M). Since (5{Mi) > 2 it follows that (i{M\) > (d{M). Remark. To see why the condition fd{M\) > 2 is necessary, let M — U\^2- T hen /3(M i) 0 and ^(Mi) /?(M) = 2 . (8 ) Let (T, I) be an optimal branch decomposition of M. Then it is also a branch decom position of iV. If e G E{T) then V'(W, r, Z, e) = cr(Ae) + o-(Bg) - (r(g(N)) 4-1 < p(Ag) -f- p(Bg) - p(5'(M)) -b 1 < 0(M ,r,/,e). Therefore, p{N) < $(W,T,Z) < ^(M,T,Z) = p{M). (9) By (4), it suffices to show that P{M) < max{/9(M • Cp : 1 < i < k}. Since the branch-width of any matroid having less than 2 elements is zero, we may, without loss of generahty, assume that every connected component of M has at least 2 elements. We prove the claim by induction. The result is true when k = 1. We assume that it is vahd for k = r — 1, where r > 2, and suppose k — r. Let M\ be the submatroid of M with connected components C j,..., Let {T\Ji) be an optimal branch decomposition of Mp Then ^(T p/i) — P{M\) = 43 max{/3(M • C,) : 1 < z < k}. Let (Trjr) be an optimal branch decomposition of M Cr. Let and be leaves of T\ and Tr, respectively, with and as incident edges and x-j and as labels, respectively. We construct a new ternary tree T from Ti and Tr, by attaching pendant edges Cj and to and respec tively, and joining the vertices n, and by an edge j. Let the pendant vertices that are adjacent to and be and respectively. To obtain a branch decomposition (T,/) of M we define I : L{T) — )■ S{M) by l{z) = li{z) , ifzeL(Ti)\ni = lr{2) , if z G L(Tr)\ur (2-9) if z = ■’I ’ if z = (See Figure 2.3). (Tr,L) [T,l) Wl Figure 2.3. Constructing (T,l) from and {Trjr) Let e E E[T). It suffices to show that = jp{M,T,l,e) < rn, where m = max{/l(M • Ci) : 1 < i < r}. Since \Ci\ > 2, whenever 1 < f < r, it follows that m > 1. There are 5 cases to be considered. case(i); e = j. T hen ^|){e) = tl>{j) = p{M\) 4 - p(C'r) — p{M) -t-l = 0 + l= :l< m . 44 case(ii): e = e,. Then ip{e) ^ ^ ( e j = /9 ({xJ) +p(5(M)\{a;i}) - p{M) + 1 = + p{S{M\)\{x^}) + p{Cr) — p{Mi) - p{Cr) + 1 ( 1 0 ) = p({^’i}) + /’('S'(^i)\{^î}) - p(Mi) + 1 = < P{Mi) < m. case(iii): e = c,.. This proof is analogous to that of (ii). case(iv); e G E{T\). Let, Ae = Ae{T,l) and Be — Be{T,l). Without loss of generality, suppose Cr Ç Ae- Then V’(e) = p{Ae) + p{Be) — p{M) + 1 = p{Cr) + p{Ae\Cr) + p{Be) — p{Cr) — p{M\) + 1 — p { ^ e \C r ) + p{Be) ~ p{M\) + 1 = 0(M i,Ti,/i,e) < (3(Mi) < m case(v): e G E{Tr). This proof is analogous to that of (iv). Remark. To see that some connected component must have at least 2 elements in order for the result to be true, consider the matroid 172, 2 - This matroid has branch-width 1 but each of its connected component submatroids has branch- width equal to 0 . ■ Since both connectivity and branch-width are defined in terms of separations, it is possible to estabhsh a relationship between these two concepts. Proposition 2.1.4. Suppose M is an n-connected matroid where n > 3. Then P{M) > n if and only if |5 (M )| > 3n — 5. Proof. If f3{M) > n then, by (3) of Proposition 2.1.3, it foUows that \S{M)\ > 3n — 5. Conversely, suppose |5'(M)| > 3n — 5. Assume (i{M) < n. Then, because [(3n — 5)/3] = n — 1, by the central-edge-lemma it foUows that M has an {n — l)-separation, and this violates the n-connectivity of M. B We find the branch-width of some specific matroids. 45 Proposition 2.1.5. (1) If a matroid M has no non-loop circuits and has at least 2 elements then its branch-width is equal to 1 . (2) If a matroid M consists of a non-loop circuit then its branch-width is equal to 2 . (3) Each of the matroids (72,4, ^{K ^), Fj, F j, and has branch-width equal to 3. (4) Each of the matroids M{K^) and M*{K^) has branch-width equal to 4. (5) If k,n G Z such that n >2 and 0 < k < n then (2-12) f^{Uk,n) = 1 + min{ fn/3], k, n — A;} . Proof. By (2 ) of Proposition 2.1.3, for each matroid M it suffices to prove the result either for M or for M*. For each matroid M being considered, let (T,/) be an arbitrary branch decomposition of M and e G E{T). (1) Since |5'(M)| > 2, it follows that /?(M) > 1. Since M has no non-loop circuits, by Proposition 1.7.7, it follows that ^(e) = < 1 . Hence, the result follows. (2) Since M consists of a circuit, p{M) = |S(M)| — 1 and if X C 5(M) then p{X) = \X\. Therefore, ip{e) = p{Ae) 4- p{Be) — p(M) - f 1 = \Ae\ + \Be\ - (|5(M )| — 1) -b 1 = 2. Hence, the result follows. (3) If M = (72,4 then the branch decomposition (T, /) is essentially unique. If c is the central edge of T then rp{c) = 2-\-2 — 2-j-l — 3. Hence, (i{U2^A) — 3. If M = M{K^) then, by (3) of Proposition 2.1.3, it follows that < 1 -f- [6/3] = 3. On the other hand, by the central-edge-lemma, 3c G E[T)such that either \Ac\ = 2 and \Bc\ = 4, or jAd = 3 and \Bc\ = 3. T hen either p(Ac) > 2 and p{Bc) = 3 , or p[Ac) — 3 and p{Bc) > 2 . Therefore, tj){c) > 3. Hence, (5{M{Kt\)) > 3. \î M = F'l then S{M) is the union of 3 circuit-hyperplanes. Therefore, by (2) of Proposition 2.1.2, it follows that fd{M) < 3. On the other hand, M{K/\) is a minor of M. Hence, by (4) of Proposition 2.1.3, it follows that P{Fj) > 0{M{K^)) = 3. Figure 2.4 illustrates a EucHdean representation and an optimal 4 6 branch decomposition of Fj. It can be verified that every pendant edge of this branch decomposition has order 2 and that every non-pendant edge has order 3. ® © Figure 2.4. A Euchdean representation and a branch decomposition of Fj If M = then S{M) is the union of three 3-cocircuits. Therefore, by (1) of Proposition 2.1.2, it foUows that < 3. On the other hand M{K 4) is a minor of Therefore, by (4) of Proposition 2.1.3, it foUows th at > 3. (4) Suppose M = M{K^). Then S{M) can be expressed as the union of a 4- circuit and two subsets of cardinaUty 3. Therefore, by (1) Proposition 2.1.2, it foUows th at (3{M{K^)) < 4. On the other hand, by the central-edge-lemma, 3c G E{T) such that \Ac\ > 4 < \Bc\. Then p{Ac) > 3 < p{Bc)- In order to show th a t P{M{K^)) > 4 it suffices to prove that either p{Ac) > 3 or p{Bc) > 3. Assume otherwise. Then p{Ac) = 3 — p{Bc)- Then both ■ Ac and A ' 5 • Be include aU the vertices of A' 5 . This is because if one of these subgraphs does not, then the other subgraph would contain aU the edges that are incident with some vertex and that subgraph would have rank 4. Because A ' 5 • Ac and A 5 • Be contain aU 5 vertices of A 5 , they must both be disconnected and have exactly 2 components each. Also, one of the components must consist of a single edge and the other can have at most 3 edges. Then |Ac| < 4 > \Bc\- This is a contradiction, because \Ac\ 4 - \Bc\ < 8 < 1 0 = |A ( 7 v5 )|. (5) Suppose M — where k and n are as given. By (3) of Proposition 2.1.3, it foUows th at /3(A/. „) < 1 -|- min{ [n/3] ,k,n — k}. 47 B ([n/3], 1 + [n/3]) (n - [n/3], 1 + [n/3]) r a n c h w i d t 1 - h (n, 1) 0 0 [n/3] Rank ^ n Figure 2.5. Rank versus branch-width of (7^. „ ,where n > 2 To prove the lower bound, we may, without loss of generahty, assume that k < n — k, because Ujt „ and Un-k,n ^.re duals of each other, and so have the same branch-width. Then k < [n/2]. By the central-edge-lemma, 3c G E{T) such th at \Ac\ > [n/3] < \Bc\. W ithout loss of generahty, let \Ac\ < \Bc\- Then \Bc\ > [n / 2 ] > k. Therefore, p{Ac) > min{ [n/3], /j, n — k} and p{Bc) = k. Then ip{c) = p{Ac) 4 - p{Bc) — k + I > min{ [n/3] ,k,n — k} + k — k + 1 . Hence, P{Uk,n) = 1 + min{ [n /3 ], fc, n - k}. ■ Figure 2.5 gives the plot of /3{Uk^n) against k, for a fixed value of n > 2. We prove a branch-width characterization for series-parallel graphs. Proposition 2 . 1 . 6 . A w,atroid M is the polygon matroid of a series-parallel graph if and only if j3[M) < 2. Proof. First suppose M is the polygon matroid of a series-parallel graph. By (9) of Proposition 2.1.3, we may assume that M is connected. If |S(M)| < 1 then (3{M) = 0, and if 15(M)j = 2 then (3{M) = 2. We proceed by induction. Assume |5'(M)| > 3. Then, M or M* is a parallel extension of a matroid M\ which is the polygon matroid of a series-parallel network such that |5'(Mi)| — |5'(Af)| — 1 . As in the proof of (7) of Proposition 2.1.3, (3{M) = j3[M\) = 2. The inequahty now foUows. Conversely suppose (3{M) < 2. We prove the result by using (4) of Proposition 2.1.3 in conjunction with Proposition 2.1.5 to exclude the relevant minors. By (3) 48 of Proposition 2.1.5, each of the matroids (72,4, M(K 4), Fj, Fj and M( kas branch-width 3, and by (4) of that result M{K^)* has branch-width 4. Hence, none of these 8 matroids is a minor of M. Because C/2 , 4 is not a minor, M is a binary matroid. Since Fj and Fj are not minors, M is a regular matroid. Because M{K^)* and M (iv 3 ,3 )* are not minors, M is a graphic matroid. Finally, since M{I\.4) is not a minor, M has to be the polygon matroid of a series-parallel graph. ■ We now address the question as to why the tree T used in the definition of branch-width is a ternary tree. We first observe that the valencies of the vertices of T should be bounded. Otherwise T may be a star. Then the order of every edge of T would be either 1 or 2 and the resulting theory would be a trivial one. A valency bound of 1 or 2 would permit only matroids with 2 elements. So the valency bound should be at least 3. If the valency bound is 3 then we may insist on ternary trees because divalent vertices serve no purpose and may be suppressed. A theory of branch-width may also be developed by taking the valency bound of the trees to be greater than 3. Such a branch-width wiU differ from ours by some constant multiple. The trees used will not be as regular as ternary trees and computations and proofs may be somewhat more complicated. 2.2. Complete-separability In this section we look at branch-width from another point of view. We in troduce the concept of complete-separability, prove that it is a reformulation of branch-width and show how it may be used to construct branch decompositions. Definition. Suppose M is a matroid, A Ç 5(M) and k is a non-negative integer. T hen A is said to be completely k-separablc (in M) if ^{M,A,B) < k, and either |A| < 1 or A = A\ Û A2 , where ^(M, A,, B,) < k and A, is a completely fc-separable proper subset of A, for 1 < / < 2. M is completely k-separable if S{M) is completely /c-separable. 49 From this definition it immediately follows that if A is completely A;-separable and k < k' then A is also completely /c'-geparable. We prove sufficient conditions for complete-separabihty. Proposition 2.2.1. Suppose M is a matroid, A Ç S{M) and k > I is an integer. Then, if either |A| < k — 1, or \A\ < 2{k — 1) and A, B) < k, then A is completely k-separable. Proof. Since the result is trivially true if A; = 1, let A; > 2. First, suppose |j4| < k — 1. We prove by induction on \A\. The result is vahd when \A\ = 0, and so we may assume that it is vahd when \A\ = p, where 1 < p < k — 2, and suppose that |A| = p + 1. Since |/1| < A — 1, from Proposition 1.7.5 it follows that ^{M,A,B) < k. Suppose x E A. Let Ai = {æ} and Ao = A\x. Then, by Proposition 1.7.5, it foUows that f{M, A\, B\) < 2 < A and ^[M,A2,Bo) < 1 + [Agi < k. The result now foUows by the complete A-separabihty of A\ and Ag. Next, suppose |A| < 2(A — 1) and f{M ,A ,B ) < A. Then 3A%, Ag Ç A such th at A = A[ Ù Ag and |A% | < A — 1 > |Ag|. By Proposition 1.7.5 it foUows th at f{M,Ai,B{) < 1 4 - |Aj| < A for 1 < Î < 2. By the first part, both A\ and Ag are completely A-separable. Hence A is also completely A-separable. B. The foUowing result estabhshes the close relationship between branch-width and complete-separabihty. Proposition 2.2.2. Suppose M is a matroid and A is a non-negative integer. Then S{M) < A if and only if M is completely k-separable. Proof. Let us begin by disposing of the case A = 0. By definition, (3{M) = 0 if and only if |S'(M)| < 1. Also, by definition, if |5(M )| < 1 then M is completely 0-separable. Conversely, suppose M is completely 0-separable. Then |5'(M)| < 1, because otherwise there is no separation (A,B) of M such that f{M, A, B) = 0, with A and B being nonempty proper subsets of 5(M ). Now let A > 1. First suppose (3{M) < A. Let (T, Z) be an optimal branch decomposition of M and g 6 E{T). Then ^(M, Ag,Bg) — ij){g) < A. So it suffices to show that Ag and 50 Bg are completely A;-separable. It suffices to prove the complete /c-separability of Ae, for any e G E{T). We proceed by induction on \Ae\- Since it is true if \Ae \ = 1, we assum e it is true if\Ae \ = p > 1 and suppose \Ae \ = p + 1. Let v be the vertex on the A-end-tree of e that is incident with e. Since \Ae\ > 2 , it follows th at V is trivalent in T. Let d and / be the other 2 edges of T that are incident with V. Without loss of generahty, let Ae — Af^ii Aj-. Now, ^{M, A^, B^) < k > ^{M, Aj,Bj). Since both A^ and Af are proper subsets of Ac, by induction it follows that they are both completely fc-separable. Therefore, Ae is completely fc-separable. Conversely, suppose M is completely fc-separable. We describe how a branch decomposition (T, 1) of M such that < fc may be constructed. By the complete fc-separabihty of M, it follows that there is a separation {Ai,A 2) of M such th at ^{M, A{, Bi) < fc and A{ is a completely fc-separable proper subset of S{M), for 1 < i < 2. At this stage Bi = A 2 and B 2 — A\. The first step of our construction is to label the two vertices v\ and ng of an edge e by A\ and Ag, respectively. Then t/)(e) — ^(M, AjjAg) < fc. At any stage of the construction if a vertex w is labelled by a non-singleton subset P of M then we consider the partition (Pj, Pg) of P where ^(M, P,, Qi) < fc and P% is a proper subset of P , as given by the complete fc-separabihty of M. Here Qi — Pf, for i = 1, 2. We attach two edges /] and /g to re, with pendant vertices w\ and n;g, respectively, and then label w\ and wg by P\ and Pg, respectively. Since i>{fi) — i{M, Pj, Q{) < fc, we wiU have a branch decomposition (T, /) of M such that ^(T, I) < fc when our construction terminates. B It is interesting to compare the above result with Proposition 2.1.1. As stated earlier. Proposition 2.1.1 is helpful in testing a given branch decomposition for a particular upper bound. We show that Proposition 2.2.2 is useful in the con struction of a branch decomposition with a particular upper bound in mind. The actual construction of a branch decomposition (T, I) of M, with \P(T, I) < fc takes place in several stages. We begin with a separation (P, Q) of M where ^{M,P,Q) < fc. (Usually it is preferable to select this first separation so that |P| > |5'(M)|/3 < \Q\, as required by the central-edge-lemma.) Our first ternary tree consists of a single edge e. We label the two vertices adjacent with e by A 51 and B, respectively. We refer to the labellings that arise during this construction as vertex-labellings by M of width < k. We proceed inductively. If at any stage a vertex v is labelled by a singleton subset {x} of M then that vertex v shall be a leaf of T with l{v) = x. On the other hand, suppose the vertex-label of v is a subset A of M where |A| > 2. Then we must find a partition , A 2 ) of A where Af is a proper subset of A and ^(M, Ai,Bi) < k, for 1 < f < 2. If we find such a partition of A, we then form a new vertex-labeUing of M by attaching two pendant edges to v and labelhng the corresponding pendant vertices with A\ and A 2 , respectively. (It is clear from the construction that all trees formed in this manner are ternary.) If no such partition of A can be found, we must abandon the present vertex-labeUing and back-track to the vertex-labeUing wliich gave rise to the present vertex-labeUing and proceed. If we obtain a vertex-labeUing where every leaf of the tree is labeUed by a singleton subset of M then by letting T be our present ternary tree and setting l{v) to be equal to the element of the singleton that labels a, for every leaf v of T we have a satisfactory branch decomposition (T, I) of M with U'(r, /) < k. In the case of a successful construction, we say that any vertex-labeUing which was used in forming (T, /) and was not discarded due to back-tracking, is extendible to a branch decomposition of M of width < k. In Figure 2.6 we use the Euchdean representation of F 7 that was given in Figure 2.4 and show how the optimal branch decomposition of F 7 that was also given in Figure 2.4 may be constructed, by means of the algorithm described above. The next result shows how a knowledge of the complete-separabihty of partic ular sets can be helpful in the construction of desirable branch decompositions. Proposition 2.2.3. Let M be a matroid and k a positive integer. If the label of every vertex of a vertex-labelling of M is completely k-separable then that vertex- labelling of M may be extended to a branch decomposition (T, /) of M such that <&. Proof. This foUows by the description of the construction of a branch decompo sition using vertex-labeUings. H 52 a b a b 1 (v) a b <5 ) (iv) (vi) F ig u re 2 .6 . Constructing an optimal branch decomposition of F 7 Using this result in conjunction with Proposition 2.2.1 it is possible to confirm an upper bound for the branch-width of a matroid without carrying out the construction of a branch decomposition completely. For example, it is possible to assert that 0{Fj) < 3 after the first vertex-labelling given in Figure 2.6. The following result is an easy corollary of Proposition 2.2.3 Proposition 2.2.4. Suppose M is a matroid and k is a positive integer. If M is the disjoint union of 3 completely k-separable sets then P{M) < k. B 2.3. Branch-width of representable matroids We prove a result about upper bounds for the branch-widths of representable m atroids. Proposition 2.3.1. Let r be a non-negative integer. Then (1) If M is the binary matroid that consists of all the vectors in the vector space {GF{2)Y then (3{M) = r; (2) If M is a binary matroid such that p{M) > 2 then ^(M) (3) Suppose r > 1 , and q ^ 2 is a prime power. If M is the representable matroid that consists of all the vectors in the vector space [GF{q)Y then /3{M) = r + 1 . Proof. We begin by stating some facts for later use. Suppose M is the binary matroid that consists of all the vectors in the vector space {GF{q)Y, where r is a positive integer and g is a prime power. If iî G then \H\ = q^~^ and if Hj G Jhf{M) such that Hj ^ Hj then Hi + Hj = {uf + Vj : Vi G Hi and vj G Hj] = S{M) and (2-13) p{HinHj) = r - 2. Also, if H\, H2, H^ £ Jf{M) then, by the inclusion-exclusion formula, (2-14) \HiUHoUH:i\ = ^ \H i\-( ^ \Hi n Hj\) + \Hi n H 2 n H^^\. i=l ^l (1) If r = 0 then M consists of a single loop and (3{M) = 0 = r, as required. If /■ = 1 then M consists of a loop and a coloop. Hence, (I{M) = 1 = r, as required. Suppose r > 2. We prove that /3{M) < r by means of Proposition 2.1.2. Without loss of generality, suppose 3H\,H 2 £ J^{M) such that Hi Y H2 and H \li H2 Y S{M). By formula (2-13), it foUows that p{H\ D H2) — r — 2. Therefore, if z £ S{M)\{Hi U H2) then H^ = {{H\ n H2) U {x}) £ Also, H\ ^ H[i Y ^ 2- Because H3 ^ fl H2, it foUows th a t H\ fl H2 fl H 3 = fl H2- Hence, by formula (2-14), it foUows that (2-15) \Hi U H 2 UH 3 I = 3(2"-^) - 3(2'--) + 2 '- ^ - 2 ’’ = \S{M)\. Therefore, S{M) = H\ U Hq U H 3 and, by (2) of Proposition 2.1.2, it foUows that /3(M) < r. In order to show that (3{M) > r, we consider an arbitrary branch decomposi tion (T, /) of M. Then, by the central-edge-lemma, 3c £ E{T) such that \Ac\ > [15(M)|/3] < |i?c|. Since [|5(M )|/3] > 2’’“^, it foUows that \Ac\ > 2 '’~^ < \Bc\. Hence, p{Ac) > r - 1 < p{Bc). It suffices to show that either Ac or Be is of fuU rank. Assumep{Ac) = r — 1 = p{Bc). T hen \Ac\ < 2’’~^ > \Bc\. Since Ac C\ Be — 0, the zero vector cannot 54 belong to both Ac and Bq. Therefore, either |i4c| < 2'’“ ^ or \Bc\ < . Hence \Ac\ + \Bc\ < |5(M )|, which is a contradiction. Therefore, (5{M) > r. (2) Suppose M is a binary matroid such that p{M) > 2. By (2) of Proposition 2.1.3, it suffices to prove that P{M) < p{M). Without loss of generahty, suppose (3{M) > 2. Let N be an underlying simple matroid of M. By (7) of Proposition 2.1.3, it follows that (3{N) = Since Æ is a submatroid of the matroid that consists of all the binary vectors of it follows, by ( 1 ), th at < p(M). Hence, f3{M) < p{M) as required. Remark. We see that (2) need not be true when p{M) = 0, by letting M consist of 2 or more loops. In this case P(M) = 1 > 0 = p{M). We also see that (2) may fail when p{M) = 1, by letting M = U\^2- this case (3{M) = 2 > 1 = p(M). (3) If r = 1 then M consists of one loop and g — 1 > 2 parallel elements and P{M) — 2, as required. So suppose r > 2. By ( 2 ) of Proposition 2.1.2, it suffices to show that S(M) cannot be expressed as the union of any 3 hyperplanes of M. We prove this fact by a cardinality argument. Let Hi, J? 2 a^nd be 3 distinct hyper planes of M. From formula (2-13) it follows that \H{ n Hjl = for 1 < i < y < 3. Also, p{HinH 2nH-i) = p{HinH 2)+p{H:i)-p{{HiriH 2)+H2,) < r - 2+r-l-{r-l) — r — 2. Therefore, |(LTi fl H2 H Hg)| < Then from formula (2-14) it follows th at \Hi U F 2 U H 3 I < 3 (/-l) - 3(/-^) + gr-2 (2-16) =3(/-^)-2(/-2) <3(g'-^) Therefore, H\ U H2 U H 3 C S{M). Hence the result follows. Remark. When r = 0, we see that (3) is false because then /3(M) = 0 also. ■ By using the central-edge-lemma we obtain an upper bound for the size of a simple matroid that is representable over a finite field, in terms of the rank and the branch-width of the matroid and the cardinaUty of the field. 55 Proposition 2.3.2. Suppose M is a simple matroid that is representable over GF{q), where q is a prime power. Suppose also that (2-17) r(M) > /3(M)-1- [logg(9/2)J, where r{M) = mm{p{M),p*{M)}. Then (2-18) | 6 '(M )| < Proof. It suffices to prove this result when r(M) = p{M). Let s = |5(M)|, r = p{M) and b = f3{M). Also, let (T,/) be an optimal branch decomposition of M. By the central-edge-lemma, 3e 6 E{T) such that \Ae\ > s/3 < \Be\- W ithout loss of generality let \Be\ > |Ae|. Then 3x, with 0 < x < s / 6 such that \Ae\ = s/3 4 - X and \Be\ = 2 s/3 — x. Since M is simple p{Ae) > log^ |Ae| = logg(s/3 -b x) and p(Bg) > log^ \Be\ = log^(2s/3 — x). Since V’(e) < 6 it follows th at p{Ae) 4 - p{Be) < r + b — l. Therefore, log^{(s/3 4 - x)(2s/3 — x)} < r + b —1 and (s/3 -f x)(2s/3 — x) < q'^'^^~^. We now consider the function (2-19) /(x) = (s/3 4- x)(2s/3 — x), where 0 < x < s/ 6 . Since its derivative is given by f'(x) = s/3 — 2x, the function / is increasing on the given interval. Therefore, in tliis interval the minimum value of /(x) 2 ^ 2 9^2 is /(O) = Hence we have that and the required inequality follows. Remark. The condition r(M) > P{M) 4 - [log,.^(9/2)J ensures th at the proved upper bound for |5'(M)| is less than q^^^\ which is an obvious upper bound. H For binary matroids we have the following immediate corollary. Proposition 2.3.3. Suppose M is a simple binary matroid with r[M) > (3{M)-\- 2, where r{M) = m.m{p{M), p*(M)}. Then (2-20) |5'(M)| < The last result of this section states that all uniform matroids are representable. Although this fact is not related to branch-width, we include it for reasons of general interest. We need the following Lemma. 56 Lemma 2.3.1. Suppose F be a field and a,,..., E F , where r E Let the (r X r) square matrix A =■ ((is,()rxr be defined by Ugy — for 1 < s,t < r- Then, (2-21) det(A) = (-l)'-('-l)/- n («.'-«(). Ks<< Proof. Let D and P denote the right and left sides, respectively, of the above equation (2 - 1 - 1 2 ). / 1 1 1 1 \ « 1 Q 2 0 3 O r “ 1 « 2 « 3 ' « r (2-22) . o 3 «:2 « 3 - 4 " ^ T reat a i,...,a r as independent variables. If a,- = aj for some i, j such that 1 5 z < j < r, then D = 0. Therefore, a, — aj is a factor of F, for 1 < i < j < r. Then the degree of either side, in aj,..., a,-, is 1 -b 2 -f • • • -|- (?■ — 1) = r{r — l)/2. Therefore, D = kP, where k E F. The coefhcient of aoa^ ■. .a^~^ in D is 1 (leading diagonal) and in P this coefficient is (—1)(^+^^ ^(>•- 1 )) _ (_ l)'’(^-l)/2. Since r(r — 1 ) is even k m aybe taken as ( —l)’’(^~l)/2. g Proposition 2.3.4. Let k,n E Z such that 0 < k Proof. Because a matroid is representable over a field if and only if its dual is also representable over that field and and „ are duals of each other, we may, without loss of generahty, assume that 0 < k < [ n / 2 ]. ( 1 ) The result is vaUd because Fo^n = M(Gq), where Gq is a graph that consists of n loops and Fi^„ = M (Gi), where G\ is a graph that consists of a set of n parallel edges. (2) Suppose 2 < k < n — 2. Let F be a field that has at least n 57 distinct elements. It suffices to construct n vectors, over such that any k of them are hnearly independent over F and any A: + 1 of them are not. Let «1 , 0 2 ,..., « 7 1 be n distinct elements of F. / 1 \ a; (2-23) Define n,- , for 1 < z < n and 5 — {v; : 1 < z < zz}. Since 5 is a subset of a A—dimensional vector space, over F, any A -|- 1 of these n vectors are linearly dependent over F. In order to show that any A of these vectors are hnearly independent, we may, without loss of generahty, let the A vectors be , U2 ,..., If A is the (r x r) square matrix which has v\^V2-, ■ ■ ■ -,vi. as its column vectors then A is the matrix defined in Lemma 2.3.1 with v = k and, by that result, det(A) ^ 0 because a; ^ Oj whenever 1 < i < j < k. Therefore, zzj, z; 2 ,... ^Vf. is hnearly independent. ■ CHAPTER III Tangle Number and a Min-max Theorem for Matroids 3.1. Matroid tangles The concept of a tangle and the associated invariant, which is called the tan gle number was first introduced for graphs by Robertson and Seymour [4], We develop these concepts for matroids. Loosely speaking this is achieved by sub stituting (matroid) rank for (graph) vertex set. In this chapter we shall use the results that were proved in Section 1.7 and the notational conventions that were introduced in that section., In this section we define tangles for matroids and estabhsh results for later use. Roughly speaking, a tangle of a matroid M is a set of low-order separations of M, where for each such separation (A,B) we select either {A,B) or (B,A), and do this in a consistent way. If .3^ is a tangle of order t > 0, then for every separation (A, B) of order less than t, the first tangle axiom states that either (A, R) £ 5" or (R, A) £ If (A, R) £ 5^ then A and R are referred to as the small side and the big side, respectively, of the separation (A, R) (with respect to to the tangle y). Some justification for this terminology may be found in the second and third tangle axioms. The second tangle axiom states that M is not the union of any 3 small sides, and the third tangle axiom states that no small side spans all of M. We now define formally. Definition. Suppose M is a matroid and f is a positive integer. A tangle of order t in M is a set y of separations of M, each of order < t such that the following axioms are satisfied. 58 59 (i) VA Ç S{M) such that ^{M,A,B) < t, either (A,B) 6 ^ or (B,A) 6 (ii) E(Ai,Bi),(A2,B2),M3,^3) E ^ then Al UA2 U As f 5'(M); (iii) If (A,B) G 5", then p{A) < p{M). We refer to these axioms as the first, second and third tangle axioms. The tangle number of M, which is denoted by 6{M), is defined as follows: If there exists a set 5" of separations of M satisfying the second and third tangle axioms and such th at VA Ç S(M) either (A,B) G ^ or {B,A) G ^ then 6{M) = oo. Otherwise, 6{M) is the maximum order of tangles in M, and 6{M) = 0 if M has no tangle. If 9{M) = oo then any set ^ of separations of M that satisfies the defining conditions for 6{M) = oo given above is called a maximal tangle of M. Otherwise, any tangle A' of M of order 6{M) is called a maximal tangle of M. If p{M) > 0 then {(0, S{M))}, which is the unique tangle of order 1, is called the trivial tangle of M . As an example we compute 0{M) where M — Fj. Because {(0, 5(M))} U {(z, y ) : z G S{M)} is a tangle of order 3 it follows that that 6{M) > 3. In order to prove that 0{M) = 3 we assume that M has a tangle ^ of order > 4. Now if A is a circuit-hyperplane of M then since ^{A,B) = 3 and B spans M it follows th at (A,B) G Then the second tangle axiom is violated. Hence, 6{M) — 3. Proposition 3.1.1. Suppose M is a matroid. Then (1) The following are equivalent: (a) 9{M) is finite; (c) M has no coloop; (2) 9{M) > Q if and only if p{M) > 0; (3) g(M) ^ 1; (4) If 9{M) is finite then M has a tangle of every positive order n < 9{M); (5) If 6{M) = oo then M has a taiigle of every positive order; 60 Proof. (1) (a)=J>(b): Suppose (a) is true. If {A,B) is a separation of M then < p{M) + 1. If B{M) > p{M) + 2 then there is a tangle 5" of M such th at VA Ç S{M) either {A, B) E ^ or (5 , A) G Therefore, 6{M) = oo, which is a contradiction. (b)=4-(c): Suppose (b) is true. Assume M has a coloop, x (say). Then since {{A,B) : A Ç S{M) and x ^ A} satisfies the second and tliird tangle axioms it follows that 6{M) = oo. This is a contradiction. (c)^(a): Suppose (c) is true. Assume 6[M) — oo. Let 5^ be a maximal tangle of M. Since £7’ satisfies the third tangle axiom it follows that p{M) > 0 and S{M) 7 ^ 0. Then (x,T) G . f , for all x G S{M), because p{Y) = p{M). Let Aq be a maximal element of {A ; {A,B) G £^}- Then by the third tangle axiom Aq § S(M). Let z G B q. Then ((Ag U z)'^,AQ U z),(Ao,Bo),(z,5'(M )\z) G 5". This violates the second tangle axiom. (2) If p{M) > 0 then 6{M) > 0 by the existence of the trivial tangle of M. Conversely, if 6{M) > 0 then since M has a tangle, by the third tangle axiom it follows that p{M) > 0. (3) Assume 9{M) = 1. T hen M has a separation of order 1 but no tangle of order 2. Suppose A Ç S{M). Then from (2) of Proposition 1.7.4 it follows that (A,B) is a separation of order 1 if and only if A is a nonempty proper subset of M which is also a union of components of M. Since 6[M) = 1 it follows that M has a component X (say) such that p{X) > 0. Let = {(A, .B) : A is a union of components of AÎ (3-1) such th at X Ç B} . Then is a tangle of M of order 2 , which is a contradiction. (4) Suppose 5^ is a maximal tangle of M. Then the set of all separations in X whose orders are less than n is a tangle of order n. (5) Suppose 0{M) = oo and t G Z+. Let 5^ be a maximal tangle of M. Then the subset of separations of 5^ aU of whose orders are less than t is a tangle of order t. » Proposition 3.1.2. Suppose M is a matroid and £7 is a tangle of order 0 in M. Then (1) iy(A,B) e .f then (B,A) ^ (2) If{A,B),{A',B') G and^{AuA',Bf]B') < 6 then (AUA',BnB') G .^. 61 Proof. (1) Since A U B = S{M)., by the second tangle axiom it follows that {B,A) ^ (2) Because AU A! \J{Br\B') — S{M), by the second tangle axiom it follows that (B n B ', A U A') ^ 3^. Hence, {A U A', B fl B') E ^ by the first tangle axiom. B Proposition 3.1.3. Suppose M is a matroid. Let x E S{M) and t E such that t < Y), and — {{A,B) : x E B and f{A,B) < t}. Then is a tangle of M, of order t. Proof. The first tangle axiom is obviously satisfied. Because x ^ A V(A, E 3^, the second tangle axiom is also satisfied. If the third tangle axiom is not satisfied then 3{A,B) E 5^ such that p{A) = p{M). T hen ((z,n = pM+xn-p(M) + i (3-2) < p{B) 4 - p{A) — p{M) 4 -1 = 4 (^,-B ) < t , which is a contradiction. B Proposition 3.1.4. Suppose M is a matroid and ^ is a set of separations of M, each of order less than t, that satisfy the first and second tangle axioms. Then ^ is a tangle if and only if 3 5 i Ç S{M) such that the following conditions are satisfied: (a) If S\ ^ % then S\ consists of a single coloop; (b) Vx E if f,{x,Y) < t then {x,Y) E 3^. Proof. First suppose is a tangle. Let 5i = {x E S{M) : (T, x) E .T}. From the definition of Si and the first tangle axiom, it follows that (b) holds. In order to show that (a) holds suppose S] ^ 0. Assume |Si| > 1. Then 3 x],X2 E Si such th at XI ÿé X2- Hence, (y i, xi ), ^ 2 ) E 3^ and since Y\\JYi — S{M) the second tangle axiom is violated. Therefore, |Si| = 1. Let Si = {x}. Since (Y, x) E 3^ hy the third tangle axiom, it follows that x must be a coloop of M. Conversely, suppose that 3Si Ç S(M) such that (a) and (b) are satisfied. Assume that 3^ is not a tangle. Then 3{A, B) E -T such that p{A) = p{M). Choose such a separation {A,B) so th at B is minimal. By the second tangle 62 axiom B ^ Let x £ B. Then from the minimality of B it follows that {A[Jx,B\x) ^ Now (,{B\x, ALi x) = p{B\x) + p{A U x) — p{M) + 1 Therefore, by the first tangle axiom {B\x,A U x) £ Since x £ B it follows th at (3-4) $(i, y ) = p(z) + p(y) - /,(M) 4-1 < XB) -f - XM) + 1< f. Hence, by the first tangle axiom either [x,Y) or {Y,x). Because AU (jB\z) Uz = S{M) from the second tangle axiom it follows that {x,Y) ^ 5'. Hence, by (b) it follows that X £ S\. Then (a) imphes that x must be a coloop. Therefore, since z ^ A it follows that p{A) < p{M). Tliis contradicts our original assumption. ■ The extreme separations of a tangle are defined by considering the small sides of the separations of that tangle. Definition. Suppose M is a matroid, is a tangle in M and (A, B) £ . Then (A, J3) is an extreme separation (in if A' = A and B' = B, Y{A',B') £ such that A Ç A' and B' Ç B. Proposition 3.1.5. Suppose M is a matroid and is a tangle of order t > 2 in M. Suppose also that [A,B) is an extreme separation in Then (1) A is a flat of M; (2) If \B\ > 2 then B contains no coloop of M . Proof. (1) We first show that B is loopless. Assume that B has a loop x. Then ^(AUz,B\z) =^(A,B). Since (A, B) is extreme, (AUz,B\z) ^ Therefore, {B\x,A U z) E Since z is a loop of M, from (1) of Proposition 1.7.4 it follows that ^(z,y) = ^(y, z) = 1 . Since by the third tangle axiom {Y,x) ^ y , it follows that (x,Y) £ This contradicts the second tangle axiom because A U (B\x) U z = S{M). Therefore, B is loopless. To continue with the proof, we consider the cases t = 2 and t > 2 separately. First suppose / = 2. Then ((A, B) = 0 or ^(A, B) = 1. If ^(A, B) = 0 then A = 0, 63 and since B is loopless A is a flat. If = 1 then p{M) = p(A) + p{B). Therefore, A is the union of components of M and hence a flat. Next suppose t > 2. Assume A is not a flat of M . Then 3x E B where x is spanned by A. Then p{A\Jx) = p{A) and p{B\x) < p{B). Hence ^(AUx, B\x) ~ ^{A,B). From the extremity of (A, 5) it follows that (B\x, AUx) G 3^. From (1) of Proposition 1.7.4 it follows that ^{x,Y) — ^{Y,x) < 2 < t. Because A spans X, by the third tangle axiom B ^ {x}. Therefore, by the extremity of (A,B) it follows that {x,Y) G 2^. This contradicts the second tangle axiom because Au(B\x)Ux = 5(M ). (2) Suppose that \B\ > 2. Assum e B has a coloop x, of M. Then p{A U x) = p{A) + 1 and p{B\x) = p{B) — 1. Therefore, ^(A U x,B\x) — ^{A,B). By the extremity of (A, J9) it follows that (B\x,A U x) E Since x is a coloop of M, from (1) of Proposition 1.7.4 it follows that ^{x,Y) — ^(P, x) = 1. Because \B\ > 2 it follows that A c y. Therefore, by the extremity of (A, B) it follows that (x,P) G This contradicts the second tangle axiom because A U {B\x) U X = S{M). B Proposition 3.1.6. Suppose M is a matroid, S' is a tangle of order t in M and (A,B) is an extreme separation in S with |A| > 1 < \B\. Then (1) ((M,A,J9) = f-l; (2) If {B\,B 2) is a separation of M ■ B then either B\ = B or B^ = B or (3-5) ((M . B, g ], g2) > min{((M - (A U Bi), A, ), ^(M - (A U ^ 2), ^ 2)}- Proof. (1) Since (A, g) G S where A 7 ^ 0 g, it follows that f > 2. If t = 2, then ^{M, A, B) = 1 as required. So suppose t > 3. By the tliird tangle axiom p{A) < p{M). Hence, 3x G g where A does not span x. Because (A, g) is extreme (A U x,g\x) ^ S. We establish that {x,Y) E S as follows. Since |g| > 2 it follows that A C y. Therefore, because (A, g) is extreme and ((M , X, y ) < 2 < t it follows that (x,Y) E S. Now, by the second tangle axiom (g\x,AUx) ^ S. Therefore, by the first tangle axiom ^(M, AUx, g \x ) >t. Then from Proposition 1.7.8 it follows that ^(M, A, g) > ^(M, A U x, B\x) — 1 > t — 1. Hence, ^(M, A, B) = t — 1. (2) Without loss of generality, suppose B\ ^ B ^ B 2- T hen g j 7 ^ 0 7 ^ gg. 64 Because (A, B) is extreme (A U (A U ^ 3^. By the second tangle axiom either {Bo, A U B\) or {Bi,A U B 2) does not belong to Without loss of generality, suppose (^ 2 , A U B\) ^ ST. Then ((M, A U B\,B 2) > t. Hence, ^(M , A U i ? i ,S 2 ) > i{M,A,B). Therefore, p(A U 5i) + p( 5 2 ) — p(M ) 4 -1 > p{A) + p{B) - p{M) + 1. Therefore, p{B 2) - p{B) + 1 > p{A) — p{A U B\) + 1. Therefore, p{B\) + p{B 2) — p{B) + 1 > p(A) + p{B\) — p(A U B\) + 1. Hence, ^{M ■ B,B \,B 2) > i{M • {AVJ B\),A,B\). Hence, the result foUows. ■ The following examples show that the conditions |A| > 1 < \B\ are indispens able. To see the need for |A| > 1, let M = U\^2 and i = 2. Then 5^ = {(0,5(M))} is the only tangle of order 2. By the third tangle axiom (0, S{M)) is an extreme separation, but ^(M, 0,5(M )) = 0 < 2 — 1 . To see the need for \B\ > 1, let M = {72,2 ^vith S{M) = (a, 6 } and t — 2. Then ^ = {(0, 5(M)), (a, 6 )} is a tangle of order 3 and (a, b) is an extreme separation, but ^(M, a, 6 ) = 1 < 3 — 1. Proposition 3,1.7. Suppose M is a w.atroid, .9' is a tangle of order t in M and {Ai,Bi) G Suppose also that (A 9 , ^8 2 ) is a separation of Ad of order less than t, with Av containing no coloop of A7. If either (i) B 2 spans B\, or (ii) Bi has no coloop of Ad and A\ spans A 2 , then (A 2 , B 2 ) G 5". Proof. We assume (A2 , .8 2 ) ^ 3^. Then by the first tangle axiom (^ 2 , A 2 ) G 3^- Let {A,B) be an extreme separation in 3^ with B 2 Q A (and B Ç A 2 ). Case (i) B 2 spans B\. Since A D B 2 it foUows th a t A spans B\. Since (B 2 , A 2 ), (A ], B i) G .9 ' it follows that t> 2 . Since (A,B) is an extreme separation in 3^, by ( 1 ) of Propo sition 3.1.5, it follows that A is a flat of Ad. Hence, A spans no element of B. Therefore, fl S = 0. Then B\ Ç A. Therefore A U Aj = S{Ad), which is a contradiction to the second tangle axiom for 3^. Case (ii) B\ has no coloop of Ad and A\ spans A 2 . Since {A\,Bi) E ^ it foUows that ^(A i,Si) < t. If we replace A\, B\, A2 and B 2 by B 2, A 2 , B\ and Ay, respectively, then we have the same hypotheses as we 65 had for the previous case. Therefore, by that case, it follows that (B ;, A%) G and this contradicts (1) of Proposition 3.1.2. ■ We next look at tangles of connected matroids. Depending on the connectiv ity of such a matroid, it is possible to construct some of its tangles solely by restricting the sizes of the small sides of separations. We rely on the following construction. Definition. Let M be a connected matroid of positive rank and n be an integer such th at 2 < n < |5'(M)|/2. Then we define the elementary pre tangle of M of order n by letting (3-6) = {{A,B)-.AC S{M) and |A| < n -2 }. When there is no risk of ambiguity, we simply write instead of ^(M). An elementary pre-tangle of M which is also a tangle of M is called an elem,entary tangle of M . From the above definition it immediately follows that ^5^2 is the trivial tan gle and that C j, for 2 < i < j < |5(M)|/2. In addition to these observations we have the following result. Proposition 3.1.8. Suppose M is a k-connected matroid, where k is an in- teger> 2. Let m = min{ [|5(M )|/2J, Æ — 1} and n be an integer such that 2 < n < m 1. Then (a) .^5^2 zs the unique tangle of M of order 1 as well as order 2; (h) satisfies the first and third tangle axioms, for a tangle of order n; (c) If M has a tangle of order n then SI — 3^ Sin', (d) Suppose n > 3. Then M has a tangle of order n if and only if |5'(M)| > 3 n -5 ; (e) Suppose \S{M)\ > 3k' — 5 where 2 < k' < k. Then 0{M) — k' if and only if M has precisely k' — 1 distinct tangles; Proof, (a) From (iv) of Proposition 1.7.9, it follows that M has no separation of order 1 . The only separations of M of order 0 are (0,5(M )) and (5'(M),0). 66 Because is the trivial tangle it is a tangle of order 1 as well as order 2. Uniqueness follows from the third tangle axiom. (b) By (a) we let n > 3. In order to verify the first tangle axiom, let (P, Q) be a separation of M of order < n. Then, by (v) of Proposition 1.7.9, it foUows that either |P| < n — 2 ov \Q\ < n —2. Therefore, satisfies the first tangle axiom for a tangle of order n. Next suppose (A,B) G Then since |j4| < n — 2 < m, by (iii) of Propo sition 1.7.9, it foUows th at A does not span M. Hence, satisfies the third tangle axiom. (c) Once again by (a), let n > 3. First suppose (P, Q) G Then ((M, P, Q) < n. Now |P | < IQI, because otherwise \Q\ < m and then, by (iii) of Proposition 1.7.9, it foUows that P spans M, which contradicts the third tangle axiom. Hence, by (v) of Proposition 1.7.9, it foUows that |P| — ^{M,P,Q) — 1 < n — 2. Therefore, (P ,Q ) e Next suppose {A,B) G ^-^n- Then, since |A| < n — 2 < ?n, it foUows that ^{M, A, B) < n. Also, from (iii) of Proposition 1.7.9, it foUows that B spans M. Hence, (A, P) G 5". (d) First suppose M has a tangle of order n. Then, by (c), that tangle must be If \S{M)\ < 3n — 5 then S{M) can be partitioned into 3 subsets A\, Ag, A 3 , each having at most n — 2 elements. Since (A,, P%) G for 1 < î < 3, the second tangle axiom is violated. Conversely, suppose |S(M)| > 3n — 5. Then because 3(n — 2) < 3n — 5, it foUows that satisfies the second tangle axiom. Hence, by (b), it foUows that is a tangle of M. (e) We begin by making some observations. From (ii) of Proposition 1.7.9, it foUows th at M has no coloop. Hence, by (1) of Proposition 3.1.1, it foUows that 9{M) is finite and M has a tangle of every positive order < 6{M). We consider the cases k' — 2 and k' > 2 separately. If 6{M) = 2 then (a) imphes that M has precisely 1 tangle. Conversely, suppose M has precisely 1 tangle. Then that tangle has to be the trivial tangle ^s®5"g, and, by (a), it foUows th at 9{M) > 2. By (iv) of Proposition 1.7.9, any 67 separation (^4, B) of M where A ^ B has order > 2. Since contains no such separation it follows that 6{M) = 2. Next suppose k' > 2. Since M is A:-coniiected, M is also fc^-connected. Because [(3fc' — 5)/2j > A;' — 1 it follows th at m = k' — 1. If 6{M) = k’ then from (4) of Proposition 3.1.1 it follows that M has a tangle of every positive order < k'. By (a) and (c), these tangles are ^ i k'- Since these tangles are distinct, M has precisely k' — 1 distinct tangles. Conversely suppose M has precisely t' — 1 distinct tangles. To see that 9[M) < k', assume otherwise. Then from (4) of Proposition 3.1.1 it follows that that M has a tangle of every positive order < A;' + l. Since • i ^ ^ k ' tangles of M but none of these has order k' + 1 it follows that M must have at least k' distinct tangles and this is a contradiction. That 9{M) > k' follows by (a) and (c), because if 9{M) < k' then M cannot have k' — 1 distinct tangles. D 3.2. A lemma about submodular functions In this section we introduce the concepts of connectivity functions, efficient sets, biases, tree-labeUings and exact tree-labeUings and then state useful results about the interplay between these concepts. Because the mathematics in this section is independent of matroids, we have not given any proofs but relied on the proofs given in the original work of Robertson and Seymour [4]. The concepts and the results of this section will help us cross over from tangles to branch decompositions and vice-versa, and prove the min-max theorem. Definition. Let F be a finite set. A connectivity function on P is a function K : 2^ — > Z such th at (i) K(A) = K(P\A),VACP; (ii) «(A u y) «(A n y) < «(%) -k «(y ), VA,y e p . Condition (i) states that K-value is invariant under the operation of comple mentation. Condition (ii) is of course, the submodular inequality. If M is a matroid, the function k : 2^^^^ — > Z defined by k{X) = p[X) + p(A ^), VA Ç S{M), is an example of a connectivity function on S{M). 68 Definition. A subset A” Ç F is efficient (with respect to the connectivity func tion k) if k(A) < 0. Abias is a set ^ of efficient sets such that (i) if A Ç F is efficient then âS contains one of A, F\A ; (ii) if A , Y,Z then X U Y U Z ^ F . A bias ^ is said to extend a set of efficient sets if æ/ Ç We are concerned with the problem of extending a given set of efficient sets, to a bias. Definition. An incidence in a tree T is a pair {v, e), where v G V{T), e G F (T ) and e is incident with v. Suppose .£/ is a set of efficient sets that are subsets of the set F. A tree-labelling over æ/ is a pair (T, a), where T is a ternary tree and a is a function from the set of all incidences in T to the set of all efficient sets of F, such that (i) Ve G F(T) with adjacent vertices u,v, say, a (u , e) = F\a{v,e); (ii) V incidence (a, e) in T such that u is a leaf, either a{v,e) = F or 3A G .(/ such th a t a{v, e) U X — F ; (iii) if a G V{T) has valency 3, and is incident with the edges e%, cv and eg, say, then a (a , e\) U a(v, eg) U a{v, eg) = F. (See Figure 3.1) A fork in T is an unordered pair {ei,eg} of distinct edges of T with a common end (the nub of the fork). A fork {ei, eg} with nub v is exact (for a) if a(u, ei) fl a(u,eg) = 0. We say that the tree-labelling (T, a) is exact if every fork of T is exact. Proposition 3.2.1. Suppose sf is a set of efficient sets. If there is a bias ex tending s/ then there is no tree-labelling over . fl Proposition 3.2.2. Suppose is a set of efficient sets. If there is a tree- labelling over £/ then there ù an exact tree-labelling over .c/, using the same tree. 69 (2 , 6) a (v ,a a (v ,c ) a(w , c) Figure 3.1. Part of a tree-labelling (T,a) Proposition 3.2.3. Let si/ be a set of efficient sets, (T, a ) be an exact tree- labelling over £/ and (u,f) be an incidence in T. Also let Tq be the component of T \ f which contains u. Then as {v, e) ranges over all incidences of T such that V is a leaf of T and v S V{Tq), the sets F\a{v,e) are m.utually disjoint and have union F\a{u, f). H Proposition 3.2.4. If there is no bias extending sJ then there is an exact tree- labelling over Æ/. O Proposition 3.2.5. The following are equivalent: (i) There is no bias extending s/; (ii) There is a tree-labelling over js/; (iii) There is an exact tree-labelling over . ■ 70 Proposition 3.2.6. If there is an exact tree-labelling over s/ then either F — or F Çl sf, or there is an exact tree-labelling (T, a ) over such that for every incidence {v,e) in T where v is a leaf, a{v,e) ^ F. B 3.3. The min-max theorem Proposition 3.3.1 (Min-max theorem). Suppose M is a matroid. Then S{M) = 6(M) if and only if M has no coloop and j3{M) ^ 1. Proof. Suppose first that P{M) — 6{M). Since /3(M) is finite, by (1) of Propo sition 3.1.1 it follows that M has no coloop. Next assume /?(M) = 1. Then every element of M must be a loop and by (2) of Proposition 3.1.1 it follows that 6{M) = 0, which is a contradiction. Conversely, suppose M has no coloop and P{M) ^ 1. Let n to be an integer > 2. Then we define k : 2'^^^^) — > Z by «(A) = f{A,B) — n y A Ç S(M) and we let .e/ = {{z} : X G S(M)}. We prove that (3{M) = 6{M) by establishing the following : ( 1 ) K is a connectivity function. (2) is a set of efficient sets (with respect to «). (3) Suppose (T, a) is an exact tree-labeUing over .a' such that oi{l,p) ^ S{M), for every incidence {l.p) in T, where I is a leaf. For any incidence (i;, e) in T, let L[v,e) denote the set of leaves of T that are vertices of the end-tree of e that does not have u as a vertex. (See Figure 3.2). Then 3 a bijection r : L{T) — > S{M) such that a{l,p) = S{M)\t{1), V incidence {l,p) in T where / is a leaf, and a(u, e) = t {L{v, e)), V incidence (u, e) in T. (4) 3 a bias ^ that extends .c/ if and only if M has a tangle ^ of order n -|-1. (5) 3 an exact tree-labelling over if and only if j3{M) < n. (6) M has a tangle of order n -f 1 if and only if n < P{M). (7) /)(M) = g(M). ( 1 ) K is a connectivity function. Since f{A,B) = f{B,A), V separation (A,B) of M, it follows that k{X) = K(%^) y X Ç 5{M). In order to verify the submodular inequality, suppose X ,Y Ç 71 S{M). It suffices to consider the 2 cases where X = 0 and X — S{M), as well as the case where both X and Y are nonempty proper subsets of S{M). In the first 2 cases k(X U F) + k(X fl F) = k(X) + k(F) and in the last case we have K(X U F) + «(X n F) =p(X U F) + p(X'= n F":) + p(X n F) + p(X'= U F':) - 2p(M) + 2 -2 » (3-7) = k(X) + n(F ), as required. (2) .c/ is a set of efficient sets (with respect to k). If z E S{M) then, by (1) of Proposition 1.7.4, it follows that k({z}) < 0. Hence, {z} is efficient. (3) Suppose (T,a) is an exact tree-labeUing over of such that a{l,p) ^ S{M), for every incidence (I,p) in T, where I is a leaf. For any incidence (n,e) in T, let L[v,e) denote the set of leaves of T that are vertices of the end-tree of e that does not have u as a vertex. (See Figure 3.2.) Then 3 a bijection r : L{T) — S{M) such that a{l,p) = 5(M )\r(/), V incidence (/,p) in T where Z is a leaf, and a(u,e) = r(jL(n,e)), V incidence {v, e) in T. L{v,e) Figure 3.2. Sets L{u,e) and L{v,e), for incidences (u,e) and (n,e) of T First suppose (Z,p) is an incidence in T, where Z is a leaf. Then by condition (i) on tree-labeUings and the fact that a(Z,p) ^ S{M), it foUows th a t 3z G S{M) such th at a{l,p) — F. We define r : L{T) — )■ S{M) by letting r(Z) = x. Then a{l,p) — S{M)\t{1), V incidence (Z,p) in T, where Z is a leaf. 72 Next suppose (u, e) is an arbitrary incidence in T. Let z E L{v,e) and d be the length of the path from v to z. We prove that r(z) G a{v,e), by induction on d. If d = 1 then, by condition (i) on tree-labeUings, a{v,e) = {r(z)}. Assume that our claim is true if d = t, where t > 1, and suppose d = ( -)-1. Let w be the other vertex that is incident with e, and / be the edge incident with w th at is also an edge of the path from w to z. Since the path from w to z is of length t, the inductive assumption imphes that r(z) 6 a{w,f). Then, by the exactness of (T, a), it follows that t(z) ^ a(in, e). Hence, by condition (i) on tree-labeUings, r(z) G a(v,e). Therefore, a{v,e) D T(L(v,e)). To show that a(v, e) Ç r(L(v, e)) we let d be the maximum length of a path from V to any leaf z of T, where z G L(v,e), and proceed by induction on d. If d — 1 then, by condition (i) on tree-labeUings, a(v,e) = {r(z)}. Assume that a(v,e) Ç r(L(v,e)) if d < t, where t > 1, and suppose d = t + 1. Let w be the other vertex that is incident with e. Since t + 1 > 2, it follows that w is trivalent. Let / and g be the other 2 edges that are incident with w. Then, by condition (i) on tree-labeUings, a{v,e) = S{M)\a{w, e), and by the exactness of (T, a) it follows that a(w,f) U a{w,g) — S(M)\a(w, e). Therefore, a{v,e) — a{w,f) U a{w,g). If z is a leaf of T such that z G L{w, f)\JL{^v, g) then the maximum length of a path from w to z is < t. Therefore, by the inductive assumption, a{w,f) Ç t(L(w,/)) and a{w,g) Ç r{L{w,g)). Since L{w^f) \JL{w,g) — L(u,e), it follows that a{v,e) Ç r(L(u,e)) as required. Lastly, we prove that r is a bijection. If |5(M )| — 2 then, by condition (i) on tree-labeUings, t is bijective. Suppose 15(M)1 > 2 . If r is not injective then 3 / ] , / 2 G L{T) such that l\ ^ and t{1\) = r ( / 2 ) = t (say). Let p\ be the edge that is incident with l\. T hen t ^ a{l\,p\). Since I2 G L{li,pi), it follows that t G t{L{1\,p\)). Therefore, t G a{h,p\), which is a contradiction. In order to prove that r is surjective, let u be a trivalent vertex of T, with incident edges C], 6 2 and 0 3 . By condition (iii) on tree-labeUings, a{v,e\) U a{v, e2)\Ja{v, eg) = 5(M ) and from this it follows that t{L{v, C])) U t(L (v , 6 2 )) U T(L(u, eg)) = S{M). Therefore, r is surjective. (4) 3 a bias gs that extends .e/ if and only if M has a tangle of order n + 1 . 73 First suppose M has a tangle 5^ of order n + 1 . Let â3 — {A : {A,B) E ^ } . We first show that ^ is a bias. In order to prove that the first condition is satisfied, suppose A Ç S{M) is efficient. Then, since n[A) < 0, it follows that ^(A,B) < n. Therefore, either [A^B) E ^ oi (B,A) E 5^. Hence, either A E ^ ox B E â8 . To verify the second condition, suppose X^Y^Z E SS. Then, by the second tangle axiom, XUVUZ ^ S{M), as required. To show that ^ extends s/, assume otherwise. Then 3x E S{M) such that x ^ A, V(A, H) E From (1) of Proposition 1.7.4 it follows that ^{Y^ x) < 2 < n. Therefore, {Y,x) E . Hence, by the third tangle axiom x must be a coloop of M and tliis is a contradiction. Conversely, suppose 3 a bias Sê that extends sé. Let = {[A,B) ; {A,B) is a separation of M such that ^{A,B) < n (3-8) and A E It suffices to prove that is a tangle and we do so by using Proposition 3.1.4. We must first show that Z satisfies the first and second tangle axioms. To ver ify the first tangle axiom suppose {A, B) is a. separation of order < n. T hen A and B are both efficient. Therefore, without loss of generality A E Hence {A,B) E S'. To verify the second tangle axiom suppose (Ai, Bi), (A 2 , Ho), (A 3 , J5 .3 ) E S. Then A%, Ag, A 3 E and, because ^ is a bias, A% U Ag U A 3 S{M). We show that S satisfies the conditions (a) and (b) of Proposition 3.1.4, with 5i = 0 . If 5i = 0 then condition (a) is immediately satisfied. To see that condition (b) is met, suppose x E S{M) = Sf. Then since {z} E sS, it follows that {z} E Also, from (1) of Projiosition 1.7.4, it follows that ^(z, Y) < 2 < n. Therefore, (z, F) E S, as required. Hence, by Proposition 3.1.4, it follows that S is a tangle. (5) 3 an exact tree-labelling over sS if and only if P{M) < n. Suppose f3{M) < n. Since /3(M) > 2 , it follows that |B(M)| > 2. Therefore, 3 a branch decomposition (T, t ) of M such that '5(T, t ) < n. We define a function Cl from the set of all incidences in T to the set of all efficient sets of S{M) by letting a{i>,e) = T{L{v,e)) for every incidence (v,e) of T. We first show that a is well-defined, by proving that a{v,e) is an efficient set. Suppose (u, e) is an incidence of T. Then since /?(M) < n, it follows that n{a{v,e)) < 0. Therefore, a(v,e) is an efficient set. 74 Next, we verify that a satisfies the 3 conditions of an exact tree-labeUing. By the definition of a, it follows that S{M) — a(u, e)Ùa(î), e). Hence the first condition is satisfied. If (r>, e) is an incidence in T and n is a leaf then a(n, e) U {r{n)} = S{M), and since {r(n)} £ s/ the second condition is satisfied. The third condition is satisfied due to the manner in which a was defined. Thus 3 an exact tree-labeUing over jz/. Conversely, suppose 3 an exact tree-labelling over æ/. Since |5(M)| > 1, it foUows th at 0 ^ S{M) ^ Therefore, by Proposition 3.2.6, there is an exact tree-labeUing (T, a) over such that for every incidence (n,e) in T where u is a leaf, a(v,e) ^ S{M). Then, let r ; L{T) — ^ S{M) be the bijection which was proved to exist in (3). It suffices to prove that (T, r) is a branch decomposition of M such that ^'(T, r) < n. Since r is a bijection, we have only to show that ^'(T, r) < n. Therefore, it suffices to show that ^(e) < n, for any edge e of T. Let e £ E{T) with incident vertices u and v. Without loss of generahty, we may let Ae = t {L{u, e)) = a(w, e) and Be = t (L(v, e)) = a(u, e). Then, since a(u, e) and a(v, e) are complementary efficient sets, i/>(e) = ^{Ae,Be) — ^(«(w, e), a(n, e)) < n, as required. (6 ) M has a tangle of order n -b 1 if and only if n < (3{M). M has a tangle of order n -f 1 <4- 3 a bias extending £/ (by (4)) (3-9) 44 ^ an exact tree-labeUing over (by Proposition 3.2.5) 44 n < /?(M) (by (5)) , as required. (7) /)(M) = g(M). Since M has no tangle of order 6{M) 4 -1 , by ( 6 ) it foUows that (3{M) < 6{M), and since M has a tangle of order 6{M), also by ( 6 ) it foUows th at 6{M) — 1 < P{M). Therefore /3{M) = 6{M). B Using the min-max theorem in conjunction with Proposition 3.1.8 we charac terize matroids with sufficiently high connectivity, and a particular branch-width, in terms of the number of tangles. 75 Proposition 3.3.2. Suppose k and k' are integers such that 2 < k' < k and M is a k-connected matroid with at least 2 elements. Them (i{M) = k' if and only if M has precisely k' — 1 distinct tangles. Proof. By (ii) of Proposition 1.7.9, it foUows th at M has no loop or coloop. Therefore since M has at least 2 elements, M has a non-loop circuit. Hence P{M) ^ 1. Therefore, by Proposition 3.3.1, it foUows that 6{M) = 0{M). So fi{M) = k' if and only if 6{M) — k'. Also M is fc^-connected. For k' — 2 the result foUows from (a) of Proposition 3.1.8. So suppose k' > 2. Let k" be the greatest integer such that 2 < k" < k' and |5(iV/)| > 3k" — 5. If k" — k' then the result foUows by (e) of Proposition 3.1.8. So suppose k" < k'. Since M is also ^''-connected, from Proposition 2.1.4 it foUows that P{M) = k" ^ k'. Therefore, 6[M) = k" and by (e) of Proposition 3.1.8 it foUows that M has precisely k" — 1 distinct tangles. So when the matroid does not have enough elements both sides of the equivalence are always false. ■ It is interesting to consider the instances k — 2 and & = 3 of the above result. Letting A; = 2 we see that a matroid with at least 2 elements is a polygon matroid of a series-paraUel network if and only if the trivial tangle is the only tangle of that matroid. Next, we let A: = A;' = 3 and suppose that the matroid M has at least 2 elements. Then the branch-width of such a 3-connected matroid M is equal to 3 if and only if the only tangles of M are the trivial tangle and the tangle with separations that have singletons for their respective smaU sides. Using Proposition 2.1.3 we obtain a result in regard to famiUar matroid con structs, that is analogous to that result. Proposition 3.3.3. Let M and N be matroids. Then: (1) IfN^M then 9{N) = 9{M); (2) Suppose either M has both loop and coloop, or neither. Then 9{M*) = g(M); (3) If M has no coloop then 0 < 9[M) < 1 + min{ |"|5(M)|/3"], p(M), p(M*)}/ (4) Suppose M\ is a minor of M that has a coloop only if M does. Then g(Mi) < g(M); 76 (5) Suppose |5(Af)| > 3 and x G S{M). Let M\ = or M\ = M". Suppose also that M\ has a coloop if and only if M does, and 9[M\ ) > 0. Then g(M) - 1 (6 ) Suppose M\ is obtained from M by deleting some of the loops and some of the coloops of M. Suppose also that M\ has a coloop if and only if M does. Then 9{M\) = 9{M); (7) Suppose M has no coloop and M\ is an underlying simple m,atroid of M . Then 9{Mi) = 9{M); (8 ) Suppose S{N) = S{M) and N has a rank function a such that (t[X) < p{X) VX Ç S{M), and a{S{M)) = p{S{M)). Suppose also that N has a coloop only if M docs. Then 9{N) < 9{M); (9) If C \,... ,Cf. are the connected components of M then 9{M) — max{0(M • C{) : 1 < i < k}. Proof. When the matroids in question satisfy the necessary and sufficient condi tions for branch-width and tangle number to be equal, as given by the min-max theorem, the result follows by Proposition 2.1.3. We prove it now when these necessary and sufficient conditions are not met. Since the hypotheses given in Proposition 2.1.3 have been weakened in some parts of this result we also include statements to cover those instances. We note that the tangle number of a matroid of branch-width less than 2, that has no coloop, is equal to zero. (1) Does not involve the min-max theorem. (2) If M has both a loop and coloop then 9{M*) = oo = 9{M) and if M has neither then S{M*) ^ ^ (3{M), and the min-max theorem appHes. (3) If (3{M) = 1 then 9{M) = 0. (4) If M has a coloop then 9{Mi) < oo — 6{M). Otherwise, if I3{M) = 1 then 0(M i) = 0 = 9{M), and if/3(Mi) = 1 then 9{Mi) = 0. (5) If M has a coloop then 9{M\) = oo = 9{M) and equality follows. Suppose M has no coloop. Then since 9{ALi) > 0 it follows that l3{Mi) 7 ^ 1 7^ ( 6 ) If M has a coloop then 9{Mi) = 0 0 = 9{M). Suppose M has no coloop. If /3{M) = 1 then 9{Mi) = 0 = 9{M). If P{Mi) = 0 then M i ^ [/o, 0 - Then 9{Mi) =: 0 = 9{M) If 15(Mi)| < 2 then either 9{Mi) = 00 = 9{M) or 9{Mx) = 77 0 = e{M). (7) Since M has no coloops Mi has none. Now P{M) — 1 if and only if M consists of loops. Hence, if (3{M) = 1 then 0{M\) = 0 = 9{M). Similarly, if !3{M\) = 1 the equality holds. (8 ) If M has a coloop then the inequality follows. Suppose otherwise. If /3(M) = 1 then the result follows as in (4). (9) The matroid M has a coloop if and only if some C{ has one, and in that case both sides of the equation are infinite. Suppose M has no coloop. If P{M) = 1 then both sides are zero. The condition |Q | > 2, for some i, of Proposition 2.1.3 can be dispensed with because if |Q | = 1, Vi, then both sides of the equation are either 0 or oo. ■ C H A P T E R IV Branch-widths of a Graph 4.1. Introduction If G is a graph then we may study the branch-width of M(G), as introduced in Chapter 2. In that definition of branch-width, orders of separations were calculated using rank. A similar invariant for G, which we denote by (3'{G), m ay be found using the same branch decompositions, but vertices are used instead of rank to specify the orders of separations. This in fact, is how the branch-width of a graph is defined by Robertson and Seymour [4]. In the next section we repeat the definition of P'{G) given by Robertson and Seymour and draw some elementary consequences. The rest of the chapter is devoted to the comparison of f3{M{G)) and P'{G). We consider only the cases where the values of these two invariants are less than or equal to 3. The results obtained in this chapter wiU be useful when we study the lower ideals of graphs that are defined by branch-width. 4.2. Definition of fi\G) and elementary consequences Definition. Let G be a graph. If G has less than 2 edges then the graphic branch-width of G is defined to be zero. Otherwise, we proceed as follows: Let (T, /) be a branch decomposition of M(G). Then Ve G E{T), let Ae{T, I), Be{T, I), Ac and Be denote the same respective subsets of 5(M(G)) = E{G), as given in the definition of the branch-width of a matroid in Section 2.1. 78 79 We now define the graphic order of the edge e, the graphic width of the branch decomposition (T, I) and the graphic hranch-width of the graph G, which are denoted by T,/, e), ^'{G,T,l) and 0'{G), respectively. ^'(G,T,Z,e) = |y(G.Ae(r,/))ny(GBe(T,mi = ('(G,G.Ae(T,Z),GBa(T,Z)); » '(G ,r, /) = max{^'(G, T, Z, e) : e € E{T)}- (4-1) P (G) = min{$ (G,T,l) : {T,l) is a branch decomposition of G} = min{max{V^^(G, T, Z, e) : e E E{T)} : (T, Z) is a branch decomposition of G} . A branch decomposition of M[G) is also referred to as a branch decomposition o /G . A branch decomposition (T, Z) such that = P'{G) is called a graphic optimal branch decomposition (of G). Also, when there is no risk of ambiguity we m ay denote 0'(G , T,Z, e) and '^'{G,T,l) by rp'{e) and $'(T, Z), respectively. Because P'{G) is defined in terms of edge-induced subgraphs of G, the value of P'{G) is unaffected by isolated vertices. If n < 2 then P'{Kn) — 0. We prove a general formula for n > 3. Proposition 4.2.1. P'{Kn) = [2n/3] Vn > 3. Proof. To prove that P'{Kn) < (2n/3), we consider a partition (Fi, bg, bg} of V{Kn), where |b)| < [n/3], for 1 < Z < 3. Let W\ — bg U bg, bbg = Vg U V\ and Ibg = Vi U bg. Then |bbf| < [2n/3]. Also, given any edge of Kn, both the vertices that are incident with that edge He in Wi, W2 or kbg. Let T^g, Sg} be a partition of E(G) into non-empty subsets so that the incident vertices of all the edges of E{ are contained in bbj-. By Proposition 1.3.2, we may construct a branch decomposition (T, Z) of Kn in which there is a trivalent vertex n of T so that the sets of T-leaves of the 3 branches at v are labeUed by E\,Eo and JSg, respectively. From this branch decomposition it follows that p'{Kn) < < [2n/3]. In order to show that P'{Kn) > [2n/3], we first prove that if (P,Q) is an arbitrary separation of Kn then either P or Q must be a connected spanning subgraph of Kn- First, suppose P does not span Kn- Then since Q has a vertex 80 that is not in P it follows that Q is a connected spanning subgraph of Kn- Next, suppose that F is disconnected. Let r and s be two vertices of Kn that he in different components of P. If a and b are distinct vertices of Kn, let denote the edge of K,i th a t joins a and 6 . Then kr,s E E{Q) and Vt E V{Kn)\{r, s} either kr,t E E{Q) or kg^t G E{Q). Therefore, Q is a connected spanning subgraph of Kn- Vp — Vq — Vr, F ig u re 4.1. Edges and end-trees of (T, Z) Suppose (T, /) is a graphic optimal branch decomposition of Kn and e G E(T). By convention, let Kn • Be be a connected spanning subgraph of Kn- Let Xe and Te denote the A-end-tree of e and the B-end-tree of e, respectively. We denote the vertices of Xe and Ye that are incident with e by Xe and tjc, respectively. Now because n > 3, it follows th at \V{Kn • Ag)| = ip'(e) < $'(T,/) < [2n/3] < n. Therefore, K,i ■ Ag is not a spanning subgraph of Kn- Let q G E{T) such that I Agi is maximum. Because Kn-Bq spans Kn and n > 3, it follows that j/g is a trivalent vertex. Let p and r be the other 2 edges of T that are adjacent with pq- We now prove that 81 Up = Vq = 3/r, as follows. Assume yp ^ yq. Then Xp = ijq and hence Ap ^ Aq, which is contrary to the choice of q. The proof of = yq is similar. (See Figure 4.1). We also see that if z E V{Kn) then z belongs to at least 2 of the subsets V(Kn-Ap), V{Kn-Aq) ajiàV [Kn ' Ar), oiV{Kn)- This is because, otherwise the vertex z would belong to exactly one of these vertex sets and then that vertex set would be V{Kn)-, which would be a contradiction. Let Z = {(z,%) : z e A: E . Ar)}}. Then by what we have just proved, \V{Kn ■ Ap)\ + \ V{Kn ■ Aq) \ + \ V{Kn ■ Ar)\ = |Z| > 2 n. Therefore one of V{Kn ■ Ap), V{Kn ■ Aq) or V{Kn ■ Ar) has at least [2n/3] of the vertices of Kn- Hence, f3'{Kn) — '5^(T, Z) > [2ra/3], as required. B We prove a basic result. Proposition 4.2.2. Let G be a graph. Then (1) If Gi is a minor of G then P'{Gi) < (i'{G); (2) If C\,... ,Cf. are the connected components of G then (3'{G) = max{/l'(Cj) : 1 < Z < k}; (3) 0<^'(G)< [(2|y(G)|)/3l; (4) If G is a non-loop circuit then j3'{G) = 2 . Proof. (1) If G\ is constructed by deleting isolated vertices of G then f3{G\) = P{G). Therefore, by induction we may assume that G\ is constructed from G by either deleting or contracting an edge X of G. We may also assume that |F7(G)| > 3 because otherwise P'{G\) = 0 < P'{G). Suppose (T,/) is a grapliic optimal branch decomposition of G and let v be the leaf of T such that l{v) = X. Also let e be the pendant edge of T that is incident with v. We construct a branch decomposition (Ti,/i) of Gi from (T,/) by deleting the leaf v along with the incident pendant edge e and then suppressing the resulting divalent vertex. (See Figure 4.2). Then (4-2) p'{Gi) < ^'(G i,Ti,/i) < <Î!'{G,T,l)=p'iG), as required. 82 ( T , 0 F igu re 4.2. Obtaining {TiJi) from (TJ) (2) Let b — max{/?^(Cj) : 1 < i < t}. Since C, is a minor of G it follows from (1) th a t l3'{{G) > b. To prove that (3'{G) > 6 we proceed by induction on t. Since the result is true if i = 1, we assume so if t = A: — 1, where k > 2, and suppose t = k. Let Gi = Ci U - ■ ■ Li Cf.^i and (TiJi) be a graphic optimal branch decomposition of G\. T hen = fi'{G\) — max{/?'(Gf) : 1 < i — 1}. Also let (2)., /jr.) be a graphic optimal branch decomposition of G^.. We obtain a branch decomposition (T,/) of G, from and (2/., as foUows. The ternary tree T is constructed by subdividing an edge of T\ and an edge of Tf. and joining the two divalent vertices by a new edge j. We define l\ : L{T) — > E{G) by letting Z(z) = fi(z), ifz€L(Ti) (4-3) = , if 0 G L[Tf.). Because G/. is disjoint from G\ it follows that 7jj'{TJ^j) = 0. Also for the same reason ^jj'{T,l,e) is bounded above by the maximum of ^ '( 2 "%, and for any edge e G E{T)\j. Hence it follows that /3'{G) < b. (3) By (2) we may assume that G is connected. If G is a simple graph then the result follows from Proposition 4.2.1 and the first part of this result. If G is not a simple graph, let Gi be an underlying simple graph of G. If fi'{Gi) — 0 then G = or G is the hnk-graph and the inequahty holds. If (3'{G\) = 1 then f3'{G) < 2 and the inequality holds. Lastly, if (i'[G\) > 2 then it can be shown that f3'[G\) = /3'{G). Now the given inequality follows from the earher argum ents. 83 (4) If G is a circuit that consists of 2 edges then (3'{G) = 2 because G has an essentially unique branch decomposition (T,/) and ^'(T ,/) = 2. Therefore, from (1) it follows that f3'{G) > 2. To show that /3'(G) < 2 let the sequence of edges of G starting from an edge Ai and going around the circuit be Ag,..., The branch decomposition (T,l) of G, where = 2 is constructed as follows. If n = 3 then we let T be the essentially unique ternary tree that has 3 leaves. If n > 3 we construct T from a path P that has n — 2 vertices by attaching 2 pendant edges to the initial and terminal vertices of P and attaching a single pendant edge to each of the internal vertices of P. Starting at one end of T, the leaves of T are labelled by A i,A 2-,-.-,An, respectively. B 4.3. Comparison of fS'{G) < 2 an d f3[M{G)) < 2 In order to formulate a prehminary conjecture we look at some examples. If G consists of 2 or more disjoint hnks then (3{M{G)) = 1 and P'{G) = 0. Therefore, (5{M{G)) > P'{G) in this case. If G consists of a hnk, with an edge (loop or hnk) incident with each of its vertices and G has no non-loop circuits then P{M{G)) — 1 and P'{G) = 2. Hence, P{M{G)) < P'{G) in this case. We prove the following. Proposition 4.3.1. If G is connected graph then P[M{G)) < P'{G). Proof. Since the result holds if |E(G)| < 2, we suppose \E{G)\ > 2. Let (T ,/) be a branch decomposition of M{G) and e G E{T). By Proposition 1.7.2, it follows that ^(M(G), T, 1, e) = ^{M{G), Ac, Be) < ^'{G,G ■ Ae,G ■ Be) — ip'{G,T,l,e). Therefore, $(M(G), T, Z) < '^'{G,T,l) and hence the result fol lows. B N o ta tio n . Suppose G is a graph. If P{M{G)) = P'{G) we denote the common value by /?(G). We estabhsh a result that characterizes low order graphic branch-width. 84 Proposition 4.3.2. Suppose G is a graph. Then ( 1 ) /3'{G) — 0 if and only if no com.ponent of G has more than 1 edge; (2 ) (3'{G) < 1 if and only if G has no non-loop circuit and every component of G has a vertex that is incident with all the edges of that component. (3) (3'{G) < 2 if and only if G is a series-parallel graph. Proof. (1) A connected graph has zero graphic branch-width if and only if it has no more than 1 edge. The result now follows from (2) of Proposition 4.2.2. (2) First, suppose/3^(G) < 1. Then from (1) and (4) of Proposition 4.2.2 it follows th at G has no non-loop circuit. Assume there is a component C of G such that no vertex of C is incident with every edge of C. Then either C has a path of length 3 or a path of length 3 with at least one of the pendant edges replaced by a loop. (See Figure 4.3). Each of these 3 graphs has graphic branch-width 2. Hence, /3'{G) > 2 which is a contradiction. 9 6 Figure 4.3. Excluded minors of G, for f3'{G) < 1 Conversely, suppose G has no non-loop circuit and every circuit of G has a vertex that is incident with all the edges of that component. By (2) of Proposi tion 4.2.2, we may, without loss of generality, assume that G is connected. We may also assume that |E(G)| > 2 . Let u be a vertex of G such that every edge is incident with v. Let A E E{G). Then A is incident with v. If A is a loop then A is incident only with v. If A is not a loop let u be the other vertex th at is incident with A. No other edge is incident with u because if so then G would have a circuit that consists of 2 edges. Therefore any separation of G has order < 1. Hence (3'{G) < 1. 85 (3) By Proposition 4.2.1, it foUows that P'{K4) = 3. Therefore, by (1) of Propo sition 4.2.2, it follows that if P'{G) < 2 then G has no minor isomorphic to K 4, and hence is a series-parallel graph. Conversely, suppose G is a series-parallel graph. By (2) of Proposition 4.2.2 we may, without loss of generality, assume that G is connected. We proceed by induction on e = \E{G)\. If e < 2 then f3'{M) < 2. Now G is either a series extension or a parallel extension of a series-parallel graph G\ such th a t |£'(G i)| — e — 1 . By induction it suffices to show that /3'{G) < max{/3^(Gi),2 }. Let {Ti,li) be a graphic optimal branch decomposition of Gp Also let A be the edge of G\ such that either an edge A' parallel to it was added or it was subdivided into 2 edges A\ and Ag, in order to construct G. Let v be the leaf of T such that /i(u) = A. We form a branch decomposition (TJ) of G by attaching 2 pendant edges to v and lab effing the new leaves by A and A% respectively, if it is a parallel extension, and by Ai and Ag, respectively, if it is a series extension. Then I) < max{4'^(Ti, ), 2} and from this it foUows th at /3'(G) We prove a useful coroUary. Proposition 4.3.3. Suppose G is a graph. Then (1) /)'(G) < 2 only i//3(M(G)) < 2. (2) If G is 2-connected then P'{G) = 2 if and onhj if j3{M{G)) = 2. Proof. ( 1 ) The result foUows from Propositions 2.1.6 and (3) of 4.3.2. (2) If /3{M{G)) = 2 then, it foUows by (1) of Proposition 2.1.5 that G has a non loop circuit. Then, it foUows by (1) and (4) of Proposition 4.2.2 that P'{G) > 2. Therefore by ( 1 ) it foUows that P'{G) = 2. Conversely, suppose P'{G) = 2 . Then, by Proposition 4.3.1, it foUows th at P{M(G)) < 2 . Since P'{G) = 2 it foUows th at \E{G)\ > 2 . Also, because G is 2-connected G has a non-loop circuit. Therefore, by Proposition 2.1.5, it foUows th a t P{M{G)) > 2. Hence the result foUows. H 86 4.4. A result for 3-connected graphs In this section we prove that a 3-connected graph G has graphic branch-width 3 if and only if M{G) has branch-width 3. We need the following lemma. Lemma 4.4.1. Suppose G is a ^-connected graph and are edge-induced subgraphs of G such that [Hi^Ho) is a separation of G. Suppose also that w(ffi) < w(ff 2 )- Then, if = 3 and ^ 3 then one of the following must occur. (1) The subgraph H\ consists of one component, H 2 consists of 2 single-edge components and ff{H i,E 2) = 4 . {See Figure 4.4(a)). (2) The subgraph H\ consists of two components having 3 vertices of attach ment each, H 2 consists of 3 single-edge components and {H\, H 2) = 6 . {See Figure 4.4(6)). IE (a) (b) Figure 4.4. Two separations {H\,H 2) of G Proof. Let = co{Hi), for 1 < i < 2. Then uj\ < wg and from Proposition 1.7.1 it follows that (4-4) ^'{H\,H2) = f{H \,H 2) + wi -b wg - 2 — 4- wg + 1 As ^'(i?i,i7g) ^ 3, it follows from Proposition 1.7.2 that f'{H \,H 2) > 4. Also, since the components oï H\ and ffg are edge-induced, it follows that (4-5) 2wi<2wg<('(.6fi,ff2) 87 Let Sj = — 2w;'. T hen > 0 and (4-6) ,^,.= ^ (|W(G,%)|-2) c(yfi) Hence, Sj is the sum of the number of vertices of attachment of the components of Hi in excess of 2. Then 2w% + = ^'{Hi,H2) — 2w2 + ^2- Therefore, 2uji + 2w2 + Si + S2 = 2^'(Hi, H2). Now, from (4-4) it follows that (4-7) 5i + S2 = 2 Therefore Si = 2 and S2 = 0, or Si = S2 — 1, since u>i < W2 - Before we proceed further, we observe the following. Suppose Ai E E(Hi) and A 2 E E(H 2) such th at Hi • Ai and H2 ■ A 2 are components of Hi and H2, respectively. Then by the 3-connectivity of G it follows that V(Hi • A i) fl V(H2 • A 2 ) — 0. Hence both incident vertices of Ai (respectively A 9 ) are vertices of (possibly distinct) components of H2 (respectively Hi), where each of the latter components has more than one edge. First, suppose Si = 2 and S2 = 0. Then each component of H2 must consist of a single edge. Therefore Hi has no single edge component and each component of Hi may have at most 4 vertices of attachment. If Hi has a component that has 4 vertices of attachment then (1) follows. Otherwise, Hi must consist of 2 components each of which has 3 vertices of at tachment. Therefore, H2 has exactly 3 edges. Let Ci and C2 be the components of Hi and E(H2) — {Ai, A 2 , A 3 }. If each edge of H2 is incident with a vertex of Cl and a vertex of C2 then (2) follows. Assume otherwise. Then without loss of generality, A% is incident only with vertices of Ci, while A 2 is incident only with vertices of C2 and A 3 is incident with a vertex of Ci as well as a vertex of Ü2- Then the incident vertices of A 3 are cut-vertices of G. This contradicts the 3-connectivity of G. (See figure 4.5(a)). Next, suppose = ^ 2 = 1- We show that this cannot happen. Now Hi has precisely one component, Ci (say) that has 3 vertices of attachment, and every other component of Hi consists of a single edge. Therefore, Hi can have at most one single-edge component. Let z = |V’(C'i) fl V(C2)\- Then 0 < z < 3. I f z = 3 then Hi = C,. Hence ^'(Hi,H 2) = 3, which is a contradiction. If z = 2 then again Hi = Q. Hence % ) — 2, which is a contrachction. If z = 1 then the vertex that is common to C\ and C2 is a cut-vertex of G. (See Figure 4.5(b)). This is a contradiction. Finally if z = 0 then G is disconnected, wliich is also a contradiction. (See Figure 4.5(c)). B Ho Ih (b) Figure 4.5. Impossible separations (Hi, Hg) of G Proposition 4.4.1. Suppose G is a 3-connected graph. Then fd'{G) = 3 if and only if P{M{G)) = 3. Proof. First, suppose P'{G) — 3. Then by Propo.sitions 4.3.1 and (1) of 4.3.3 it follows that /3(M(G)) — 3. Conversely, suppose P{M{G)) = 3. Figure 4.6 illustrates all the 3-connected graphs which have at most 9 elements and whose polygon matroids have branch- width 3. It can be verified that all these graphs have graphic branch-width 3. So suppose \E{G)\ > 1 0 . Assum e (3'{G) ^ 3. T hen since G is connected (i\G) > 3. Suppose (T ,/) is a graphic optimal branch decomposition of G. Then > 3. For the purposes of this proof, an edge e of T such that 0^(e) > 3, is called a high edge of (T,/). We may, without loss of generality, suppose that (T, f) has the minimum number of high edges among all the graphic optimal branch decompositions of 89 lU Prism /vs-cd ge ^ 3,3 Figure 4.6. Graphs G with |F(G)| < 9 and (3{M{G)) = f3’{G) = 3 G. Our strategy is to obtain a contradiction by constructing a graphic optimal branch decomposition of G from (T, /) such that (Ti, /i) has strictly fewer- high edges than {TJ) has. Again for the purposes of this proof, we say that such a grapliic optimal branch decomposition {T\Ji) is better th an {TJ). In all our constructions, if e G E{T) D E{T\) then 0(G,Ti,/i,e) — ip{G,TJ,e) and V,'(G,Ti,Zi,e) =V'(G,r,Z,e). Suppose e is a high edge of (T, /). We estabhsh some notation. Let G\ = G- Ag and Go — G ■ Bg. Also let caj = eo{G\) and ujo — u){G 2). Without loss of generahty, let W) < u>2. Let the vertex of the A-end-tree of e and the .B-end-tree, th a t is incident with e, be a and Z», respectively. We consider the 2 possibihties given by Lemma 4.4.1 separately. First, suppose u>i = 1, co2 — \E{G2)\ = 2 and xl)'{e) = ^ '(G i,G 2 ) = 4. Let E{G2) — {X,Y} with {z, z'} and {y,y'} being the pairs of vertices that are incident with X and Y, respectively. Then V(Gi) n y(Gg) = {z, z', y, y'}. Now, each of the two branches at b away from e consists of a single pendant edge and the two T-leaves are labelled by X and Y, respectively. Let the set of labels of the T-leaves of each of the branches at a away from e, be denoted by E{H) and E{K), respectively, where H and K are the corresponding edge- induced subgraphs of G. (See Figure 4.7). We begin by resolving the case where both H and K have 3 vertices of at tachment each. Let n — iV’( iî) fl y(Ji)|. Then 0 < n < 3. Now n 3, because otherwise 4 vertices are not available to attach X and Y. Also n 0, because otherwise the subgraph H U K has 6 vertices of attachment although the sub graph G • {X, Y} has only 4 vertices. Therefore 1 < n < 2. If n = 1 let z denote 90 G, G2 y (a) (b) Figure 4.7. Graph G and grapliic optimal branch decomposition (T, /) the common vertex of H and K. The instances n = 2 and n = \ are illustrated by Figure 4.8(a) and Figure 4.8(b), respectively. X X X X 2/ (a) n = 2 (b) n = 1 Figure 4.8. The instances n = 2 and n = \ In both these instances, we construct from (T,/) as follows. We first delete the branch at a that contains e as an edge. Thereafter, we subdivide one of the 2 edges incident with a, attach pendant edges at the divalent vertices and label the respective leaves by X and Y. (See Figure 4.9). From the separations 91 displayed in Figure 4.8, it follows that (Ti,/i) is better than (T,/) when n = I and n = 2 . Figure 4.9. Construction of (Ti,l\) From Lemma 4.4.1, it follows that if P is an edge-induced subgraph of G such that ^(P,P) = 3 and ^'(P,P) ^ 3, then either |P(P)| < 3 or |P(P)| < 3. Therefore, if \E{H)\ > 4 < \E{K)\ then = 3 = ^'{K ,K ), and we have already considered this instance. So suppose \E{H)\ < 3. Then since |P(G)| > 10 it follows that |P(A')| > 4, Also, since E{H) 7 ^ 0 it follows th a t 1(J5'(A’’))'^| > 3. Therefore, by Lemma 4.4.1 it follows that ^'{K ,K ) = 3 unless |(A(A’))‘^| = 3. The instance where |(A(A”))‘^| = 3 will be considered later, when we discuss the second possibihty given by Lemma4.4.1. Therefore, we have that \E{H)\ < 3 and ^'(K ,K ) = 3. We now consider the cases where < 3 and > 3, separately. If H has at most 3 vertices of attachment we show that \E{H)\ > 2 by assuming otherwise. Because G is 3-connected x, x', yand y' have valency at least 3 each, in G. Therefore, an edge of K is incident with each of these 4 vertices. Hence K has at least 4 vertices of attachment, which is a contradiction. Therefore, \E{H)\ > 2 < \E{H)\. Hence = 3 = (.'{K,K), a case which we have already considered. If H has more than 3 vertices of attachment then, by Lemma 4.4.1, either H has 2 edges and 4 vertices of attachment, or H has 3 edges and 6 vertices of attachment. Then H has at least 4 vertices and every vertex of P is a vertex of attachment of H. Also all 4 vertices of G • {W, Y} are vertices of attachment of that subgraph. Therefore, since K has exactly 3 vertices of attachment, it follows that G must have a divalent vertex. This is a contradiction. 92 Next, suppose ui — 2, and write C{G\) — {P,Q}. The components P and Q have 3 vertices of attachment each, and G2 has 3 single edge components. Let E{G 2 ) = {X,Y,Z} with {z,z'}, {y-,y'} and {z, z'} being the pairs of vertices of G that are incident with X , Y and Z, respectively. Then V{G\) fl V{G2 ) = {z, z ', y, rj', z, z'}. (See Figure 4.10(a)). The branches at b away from e have one and two T-leaves, respectively. W ith out loss of generahty, let the two T-leaves that are vertices of the same branch be labelled by X and T, respectively. Also, let the set of labels of the T-leaves of each of the branches at a away from e be denoted by E{H) and E{K), respec tively, where H and K are the corresponding edge-induced subgraphs of G. (See Figure 4.10(b)). Z % x' (b) Figure 4.10. Graph G and graphic optimal branch decomposition (T,/) Without loss of generahty, let \E{H)\ < \E{K)\. T hen since |P (G )| > 10 it foUows th a t |£'(7i)| > 4 < |(P(7v ))^|. Therefore, by Lem m a 4.4.1 it foUows th at K has exactly 3 vertices of attachment. We now consider the 2 cases where the number of vertices of attachment of H is equal to 3, and unequal to 3, respectively. Suppose ^'{H, P ) = 3. Since G- {AT, Y,Z] has exactly 6 vertices of attachment it foUows th at HUK must have exactly 6 vertices of attachment also. Therefore, V{H n K ) = 0. (See Figure 4.11(a)). 93 We construct from (T,/) as follows. We first delete the branch at a that contains e as an edge. Thereafter we subdivide both edges that are incident with a and attach pendant edges at the 3 divalent vertices. Finally, we label the 3 new leaves by X , Y and Z, respectively. (See Figure 4.11(b)). ^ X x' 0 0 0 (a) Figure 4.11. The graph G and branch decomposition (T,/) From the separations displayed in Figure 4.11, it follows that (Ti, l\) is better than {TJ). Finally, suppose ^'{H ,H ) J 3. Since G is 3-connected, \E{H)\ > 1. T here fore, by Lemma 4.4.1, either |J3(lf )| = 2 and H has 4 vertices of attachment or \E{H)\ = 3 and H has 6 vertices of attachment. Both these possibihties are excluded by the 3-connectivity of G. B C H A P T E R V The Lower Ideals Determined by Branch-width < 2 5.1. Introduction We begin with some notation. N o ta tio n . Suppose a is a non-negative integer. Then let % and 0 ^ denote the classes of all graphs G such that (3{M{G)) < n and (3'{G) < n, respectively. Also let and denote the classes of all regular matroids and binary matroids M, respectively, such that /3(M) < n. From the above notation it immediately follows that if m < n then — '^ni and Definition. A graph G that is an obstacle to % or is also called an obstacle to branch-width n or graphic branch-width n, respectively. Likewise, a binary m atroid M that is an obstacle to is also called an obstacle to branch-width n. In the next 3 sections we study the internal as well as the external structure of the lower ideals % and for n = 0,1,2, respectively. In Section 5.5 we study the lower ideals SSn, with respect to some summing operations. As a result of our work in this chapter, we shall be able to reduce the problems in the lower ideals % , and ^ 3 to those of 3-connected graphs and binary matroids, respectively. 5.2. The lower ideals and We define 2 summing operations on the class of all graphs. 94 95 Definition. Suppose G\ and G2 are disjoint non-null graphs. Here disjoirit imphes disjoint vertex-sets and disjoint edge-sets. The graph G\ U Go is defined to be the unique 0-sum of G\ and G 2 . The 0-sum of G\ and Gg is called the o' -sum of Gi and G2 if either G\ or G2 is edgeless. We prove some results about the 0 -sum. Proposition 5.2.1. Suppose G is the 0 -sum of G\ and Go- Then G\ and Go are proper minors of G. Also, /3'(G) = max{/ 3 '(G i),/ 3 '(G 2 )}, and /)(M(G)) =max{l,^(M(Gi)),;9(M(G2))}, i/E(G%) f 0 9 6 ^ ( 6 2 ). Proof. The proper minor condition follows because because Gj ^ fi 7 ^ G 2 . The two expressions for (d'{G) and /3{M{G)) follow from (2) of Proposition 4.2.2 and (9) of Proposition 2.1.3, respectively. In the expression for (i{M{G)), the integer 1 is included in the set of which the maximum is taken, in order to account for the instance where |£^(Gi)| = 1 = |T7(G 2 )|. B Proposition 5.2.2. A graph G is the 0 -sum of 2 of its proper minors if and only if it is disconnected. B Proposition 5.2.3. (1) The lower ideals and are finitely generated via the O' -sum arid the 0 -sum, respectively. Both a*o and have the same basis that consists of the 4 connected graphs, each of which has at most one edge. The only inactive generator for both % and S7q, is the em.pty grajrh; (2) The set of obstacles for îfg consists of the 7 graphs, each of which has no isolated vertices and exactly 2 edges. The set of obstacles for consists of the 4 connected graphs, each of which has exactly 2 edges. {See Figure 5.1 and Figure 5.2). 96 O'-sum 0 Figure 5.1. The lower ideal Sfg 0-sum Figure 5.2. The lower ideal S?q Proof. (1) From the definition of the branch-width of a matroid it follows that G E Sfq if and only if G has at most one edge. The statement about the obstacles to follows from this. By Proposition 5.2.1, the O' -sum is a proper closed summing operation on Also, from Proposition 5.2.2, it follows that every graph in Sfg is either a generator of or is a 0 -sum of 2 of its proper minors, which are also in 0 Q. Now the statement about the generation of 6% follows by induction. (2) From (1) of Proposition 4.3.2, it follows that G S if and only if every component of G has at most one edge. The statement about the obstacles to (ifg follows from this. By Proposition 5.2.1, the 0-sum is a proper closed summing operation on S^q. Also, from Proposition 5.2.2, it follows that every graph in S7q is either a generator of 0 g or is a 0 -sum of 2 of its proper minors, which are also in Now the statement about the generation of follows by induction. B 0 - 97 5.3. The lower ideals and Sfj Once again, we define 2 summing operations on the class of ail graphs. Definition. Suppose G\ and G2 are disjoint non-null graphs. Suppose also that V{ is a non-isolated vertex of G{, for i = 1 , 2 . Then the graph G obtained from G\ and G2 by identifying the pair of vertices {vi,V2 } is said to he a l-sum of G\ and (? 2 - Such a 1 -sum of G\ and G2 is called a l'-sum of G\ and Gg if a; is incident with every edge of G\ and ng is incident with every edge of Gg. (See Figure 5.3, for an example of a 1 -sum). Figure 5.3. The graph G is a 1 -sum of G\ and Gg We prove some results about 1 -sums and l' -sums. Proposition 5.3.1. (1) Suppose G is a 1 -sum of G\ and Gg. Then G\ and Gg are proper minors of G. Also, /)(M(G)) = max{l,,9(M(Gi)),;9(M(G2))},and < max{2,/3'(Gi),/(Gg)}; (2) If G is a -sum of G\ and Gg then 0'{G) = max{l,/3^(Gi),/3^Gg)} < 2. Proof. (1) We use the notation given in the definition of a 1-sum. The proper minor condition follows from the fact that v\ and ug are non-isolated vertices of G\ and Gg, respectively. The expression for f3{M{G)) follows directly from (9) of Proposition 2.1.3. The integer 1 is included in the set of which the maximum is taken, in order to account for the instance where |E(Gi)| = 1 = |F)(Gg). 98 To prove the inequality for j3'{G), let Ai be an edge of G'f that is incident with the vertex for i = 1, 2. Now suppose (Tj, Zj) is a graphic optimal branch decomposition for Gi- Let pi be the pendant edge of T; whose incident leaf is labelled by Ai- We form the ternary tree T from T\ and To by subdividing the pendant edges p[ and po and joining the resulting divalent vertices. The labelling I is defined by retaining the labels inherited from the labellings l\ and l2 - It can now be verified that ^'{G,T,l) satisfies the claimed inequahty for /3'{G). Hence the result follows . (2 ) T h at 13'{G) < m a x { l,/ 3 ^(Gi),/l^(G 2 )} follows from the construction in the preceding argument. The reverse inequahty follows from (1) of Proposition 4.2.2. Since G{ has a vertex with which every edge of G{ is incident, it fohows that Gf is a series-parahel graph, for i = 1, 2. Then, from (3) of Proposition 4.3.2 it follows th at [3'{Gi) < 2 . Therefore f3'{G) < 2 . ■ Proposition 5.3.2. Suppose G is a connected graph. Then G is a 1 -sum of 2 of its proper minors if and only if G is not 2-connected. Proof. First, suppose G is a 1 -sum of 2 of its proper minors G\ and Gg. We use the notation given in the definition of a 1 -sum . Let v be the vertex of G formed by the identification of the pair of vertices {u], ug}. Then is a cut-vertex of G. Hence G is not 2 -connected. Conversely, suppose G is not 2-connected. Let v be a cut-vertex of G. Let G\ be a bridge of G[u]. Also let Gg be a subgraph of G that is edge-disjoint from G\ and such that Gi U Gg = G and P(Gi U Gg) = v. T hen G\ and Gg are proper minors of G and G is a 1 -sum of Gi and Gg. ■ Proposition 5.3.3. (1) The lower ideal ^â\ is finitely generated via the 0-sum, and the 1 -sum. The basis consists of the 4 connected graphs that have at most one edge each. The null graph is the only inactive generator for both summing operations. The single vertex-graph is an inactive generator for the 1 -sum. {See Figure 5.4); (2) The graph that consists of two parallel edges is the unique obstacle (up to isomorphism) to the lower ideal {See Figure 5.4); 99 (3) The lower ideal 27j is finitely generated via the 0 -sum and the l' -sum. The basis consists of the 4 connected graphs that have at most one edge each. The null graph is the only inactive generator for both summing operations. The single vertex-graph is an inactive generator for the l ' - sum. {See Figure 5.5); (4) The graph that consists of two parallel edges and the graph that consists of a path of lejigth 3, as well as the 2 graphs obtained from the latter by replacmg one or both of the pendant edges of that graph by loops, are the only obstacles (up to isomorphism) to the lower ideal {See Figure 5.5). 0- and 1 -sums Figure 5.4. The lower ideal 0- and 1' -sums o o Figure 5.5. The lower ideal Proof. Parts (1) and (2) foUow from Propositions 2.1.5, 2.1.3, 5.2.1, 5.2.2, 5.3.1 and 5.3.2. Parts (3) and (4) follow from Propositions 4.3.2, 4.2.2, 5.2.1, 5.2.2, 5.3.1 and 5.3.2. B 100 5.4. The lower ideal % From Proposition 4.3.3 it follows that % = We define yet another summing operation on the class of all graphs. Definition. Suppose G\ and Gg are disjoint graphs, each of which has at least 3 edges. Suppose also that A\ and Ag are non-loop, non-isthmus edges of G\ and Gg, respectively, with incident vertex-sets {u%, and {ug,ug}, respectively. We form the graph G from G\ and Gg by identifying the pair of vertices {ui,ug} and the pair of vertices {^;i, ug}, and then deleting the edges A\ and .4g. Then G is called a 2-sum of G\ and Gg. (See Figure 5.6). Remark. There are two ways in which the vertex-pairs in a 2 -sum may be iden tified. Hence, in general there are two possible 2 -sum s of G\ and Gg. Ai A2 Gi G2 Figure 5.6. The graph G is a 2 -sum of G\ and Gg We prove some results about the 2 -sum. Proposition 5.4.1. Suppose G% and Gg are disjoint non-null graphs. Suppose also that G is a 2 -sum of G\ and Gg. Then |y(G)| = |y(Gi)| + MGg)|-2 (5-3) |E(G)| = |E(Gi)|-b|E(Gg)|-2. m Proposition 5.4.2. Suppose G is a 2-sum of G\ and Gg. Then G\ and Gg are proper minors of G. Also, ^(M(G)) =max{^(M(Gi)),/)(M(G 2))} (5-4) ;8'(G)=max{,9'(Gi),/3'(G2)}. 101 Proof. Throughout, we use the notation of the definition of a 2 -sum. The proper minor conditions hold due to the pre-condition imposed by the definition. From (4) of Proposition 2.1.3 and (1) of Proposition 4.2.2 it follows that each of the left hand sides is not less than the corresponding right hand side. To prove the reverse inequality for branch-width, let (T^, and (T 2 , Z2 ) be optimal branch decompositions of G\ and Gg, respectively. We form a branch decomposition (T,/) of G, from and (Tg, ^ 2 ), by deleting the leaves of Ti and T‘2 that are labelled by j4i and Ao, respectively, along with the corresponding pendant edges, and then joining the resulting divalent vertices by an edge. Then since '5 (r,/) satisfies the reverse inequality, the result follows. The result for graphic branch-width follows by a similar argument. B The following result will, in turn, help us to prove that % is a finitely generated lower ideal. Proposition 5.4.3. Suppose. G is a 2 -connected graph. Then G is a 2 -sum of 2 of its proper minors if and only if G is not 3-connected. Proof. If G is a 2 -sum of 2 of its proper minors then by the definition of a 2 -sum it follows that G is not 3-connected. Conversely, suppose G is not 3-connected. Then G has a separation (G^, G'2 ) of order 2, such that |£'(Gj)l > 2 < \E{G'2 )\. Let V{G'i) n y (G 9 ) = {a,n} and A( G E{G'fj, for i = 1,2. Because G is 2- connected there is a circuit G of G such that Ap Ag G E{C). Now G F (G), and G consists of a path Pi from w to a w ith A\ G E{P\) and a path Pg from u to V with Ag G P(Pg). We construct the proper minor G{ of G, by deleting all the edges of E{G'-)\E{Pi), all resulting isolated vertices and contracting all the edges of Pj except A,;. Then G is a 2-sum of G\ and Gg. (See Figure 5.7). B Proposition 5.4.4. Suppose G is an obstacle to % or where n > 2 . Ther G is 3-connected. Proof. This follows from Propositions 5.2.1, 5.2.2, 5.3.1, 5.3.2, 5.4.2 and 5.4.3. B 102 G Go G\ Figure 5.7. Obtaining two 2 -sum m ands G\ and Go of G Proposition 5.4.5. Suppose G is a 2-connected graph in Then G is a 2 -sum of 2 graphs in % if and only if |F(G)| > 4. Proof. First, suppose G is a 2-sum of Gj and G 2 , say. Then |F(Gi)| > 3 < \E{Go)\. Now, from Proposition 5.4.1, it follows that |F(G)| > 4. Conversely, suppose |F(G)| > 4. Then, by the central-edge-lemma, G has a separation {G[, G'2 ) of order 2 such that |F(G j)| > 2 < |F(G 2 )|. Therefore, G is not 3-connected. Then, by Proposition 5.4.3, it foUows that G is a 2 -sum of 2 of its proper minors. Hence the result foUows. B Proposition 5.4.6. (1) The lower ideal % is finitely generated via the 0 -sum, 1 -sum and the 2 - sum. The basis consists of the em.pty graph, the single vertex-graph, the loop-graph, the Hnk-graph, the graph that consists of two parallel edges, the graph that consists of three parallel edges and the graph that consists of a triangle. The empty graph is dormant for all the summing operations, while the single vertex-graph is dormant for all but the 0 -sum and the three other generators that have at most 2 edges are dormant only for the 2 -sum; (2) The graph K 4 is the uniqiie obstacle to % ( w isomorphism). [See Figure 5.8). 103 0-, 1- and 2-sums Figure 5.8. The lower ideal Proof. From Propositions 5.2.1, 5.3.1 and 5.4.2 it follows that the 0-sum, 1 -sum and the 2-sum are proper closed summing operations on To show that all of % is generated as claimed, suppose G E By induction, it suffices to show that either G is a generator or G is a 0 -sum, 1 -sum or 2 -sum. If G is not connected then G is a 0 -sum, and if G is connected but not 2-connected then G is a 1 -sum. So, suppose G is 2-connected. If G has less than 4 edges then G is a generator. On the other hand, if G has at least 4 edges then from Proposition 5.4.5 it follows that G is a 2-sum of 2 graphs in %, as required. The statements about inactive generators follow directly from the definitions of the summing operations. T h at K 4 is the unique obstacle to % (up to isomorphism) follows from (3) of Proposition 4.3.2. ■ We conclude this section with a result about % and 0^. Proposition 5.4.7. % = Sfg. Proof. By (1) of Proposition 4.3.3, it suffices to prove that P'{G) = 3 if and only if /3(M(G)) = 3. If G is 3-connected then this equivalence follows from Proposition 4.4.1. On the other hand, if G is not 3-connected then, by Propositions 5.2.2, 5.3.2 and 5.4.3 it follows that G is a 0-, 1- or 2-sum of 2 proper minors Gj and G'2 of G, where G%, G 2 E % or G%, G 2 G if G E % or G G 6^3, respectively. The result now follows by induction and Propositions 5.2.1, 5.3.1 and 5.4.2. B 104 5.5. One- and two-summings of binary matroids In this section we discuss two of the three binary matroid summing operations introduced by Seymour [5]. We prove that if n > 2 then SSn is closed under these two operations. Results proved in [5] are quoted without proof. Some of these results deal with the correlation between matroid connection and matroid decomposition, as apphes to binary matroids. We also include the definition of a 3-sum although we shall not discuss it here. Definition. Let M\ and M 2 be binary matroids with element sets and S 2 , respectively. We define the new binary matroid Mi AM 2 to be the matroid with element set 5iA52 and with cycles all subsets of 5']A52 of the form C 1 AC 2 , where C{ is a cycle of Mj, for i = 1,2. We are only concerned with three special cases of this operation, as follows. (i) When 5i n 52 = 0 and |5i| < |5iA52| > |52| (that is 5i ^ 0 ^ 5 2 ), M\AM ‘2 is a 1-sum of M% and M 2 . (ii) When |5i D 521 — 1 and 5% H 52 == {z}, say, and z is not a loop or a coloop of Ml or M 2 , and |5i| < |5iA52| > |52| (that is |5i| > 3 < |52|), M 1 AM 2 is a 2 -sum of Mi and M 2 . (iii) W hen |5 i H 52| = 3 and 5 i fl 52 = Z, say, and Z is a circuit of M l and Mo, and Z includes no cocircuit of either Mi or M 2 , and |5iI < |5iA52| > 1521 (that is |5i| > 7 < |52|), M \AM o is a 3-sum of Ml and M 2 . We prove a useful lemma about the circuits and rank function of a 2-sum. Here and elsewhere, p; stands for the rank function of the matroid M{ {i — 1,2). Lemma 5.5.1. Suppose M \ and M 2 are binary matroids such that 5 i n 5 2 = {z}. Suppose also that M is a 2-sum of M \ and M 2 . Then: (1) If C £ "if(M;) such that z , where 1 < i < 2, then C E tS{M). (2) If C £ if(M ) and C Ç Si, where 1 < i <2, then C £ %'(M,). (3) If C £ ‘é’(M) such that C C\ S{ ^ 0, for i = 1,2 then 3C,; £ %f(M;), for i = 1,2 such that C = C'iAC'2 , where C j fl C 2 7 ^ 0; 105 (4) 1} z ^ X i Ç Si, where 1 < i <2, then p{Xi) — pi{Xi); (5) Suppose z Ç: X i Ç. Si, where 1 < i < 2. Then p[XiAS 2 ) = P\{X\) + P2 {M2 ) - 1 and p{X 2 A S i) = P2 {X 2 ) + Pl{Mi) — 1 . Proof. (1) By taking one cycle of the symmetric difference to consist of C and the other to be the null cycle, we see that C is also a cycle of M. Since C consists of exactly one circuit of Mi, it follows that C £ ‘5f(M). (2) By symmetry, it is enough to prove the result for i — 1. Since C is also a cycle of M, it follows that C = C \A C 2 , where C\ and C2 are cycles of M\ and M 2 , respectively. Then, C2 has no circuits of M 2 because otherwise, since z is not a loop of M 2 , it follows that C must contain an element of S2 - Hence, C is the union of a set of pairwise disjoint circuits of M p Since C G V{M) there can be exactly one such circuit. Therefore, C G ^é’{M\). (3) Since C is a cycle of M and C 0 Si ^ 0, for i = 1,2, it follows that there is a non-empty cycle Wi of M%, for z = 1,2 such that C = kKjAHb- H suffices to show that each of W\ and IV2 consists of a single circuit, and by symmetry it is enough to prove this only for W i. Assum e Wi consists of at least 2 circuits. Let Ui,V\ G '«f(Mi) such that Ui r\V\ — 0 and U\,V\ C W i. Without loss of generality, we may suppose that z ^ U\. Then by (1) it follows that U\ G %'(M). Also Ui Ç C, and since U\ S\ ^ C it follows that U\ C C. This is a contradiction. (4) Let Bi be a base for M, • A,. It suffices to show that Bi is also a base for M ■ Xi- In order to show that Bi is an independent subset of M ■ X i, assum e otherwise. Then 3C G %f(M) such th at C Ç Bi. Therefore, by (2) it follows that C G 'if(M; ■ Xi). This is a contradiction to Bi being a base for M, • Xi- Next, suppose X G X i\B i. Since Bi is a base for M, • Xi, it follows that 3C G ^{M i) such th a t X G C Ç Bi \J X. Also since z ^ X i it follows that z ^ C . T hen (1) implies that C G V[M). Therefore, Bi spans M • Xi. (5) By symmetry, it suffices to prove this for i = 1, i.e., to prove that if z £ X i Ç 5i then = pi(Xi) + P 2 (-^ 2 ) ~ 1- Since z is not a loop of M\, it follows that z is not a loop of Mi • X\ either. Therefore, there is a base B\ of M \ • X \ such that z G B\. Since z is not a coloop of M 2 , there is a base B 2 of M 2 such that z ^ B 2 - Since |(Bi\z) U H 2 I = |Hi| -f IB2 I — 1, it suffices to show th at B = (jBi\ z) U B 2 is a base of M • (%% A 5 "2 ). 106 In order to show that B is an independent set, we assume otherwise. Then 3C G V{M) such that C Ç. B. We wish to apply (3). To show that C C\ 0, for 1 < i < 2 ^ suppose otherwise. Then either C Q B \\z ov C Ç B 2 an d then from (2) it follows that either C G "^{M\ • Xi) or C G respectively. This is a contradiction. By (3), it follows that 30, G for z = 1,2 such that C — C 1 AC 2 , where C\ D C 2 ^ 0. Since 2 G Si, it follows that Ci Ç Sp This is a contradiction. We prove that B spans M • (X i AS'2 ) by showing that B U x contains a circuit of M ■ (X jA S'2 ), if z G (XiA52)\S. First, suppose x G X i\Si. Since B\ is a base of M \ • X% it follows that 3C\ G V{M ■ X%) such that z G C i Ç S% U z. If 2 ^ Cl then Cl G M ■ (X i AS'2 ) an d z G C i Ç S U z, as required. If 2 G Ci then we argue as follows. Since 2 ^ S 2 , it follows that 3C2 G %?(M2 ) such that 2 G C 2 Ç S 2 U 2 . Then Cf G for z = 1,2 and Ci H C 2 = {z}. Therefore z G C 1 AC 2 Ç S U z. Also, C 1 AC 2 is a cycle of M. Therefore, 3C G V{M) such th a t z G C Ç S U z, as required. Next, suppose z G S2 \B 2 - Then 3 C 2 G V{M 2 ) such that z G C 2 Ç B 2 U z. T hen z G C 2 Ç B U z and since 2 ^ B 2 , by (1), it follows that C 2 G ’zf(M’ ■ (Xi A5'2), as required. ■ Proposition 5.5.1. Suppose M \,M 2 G ^n> where n is a positive integer. Then: (1) If M is a 1-sum of Mi and M 2 , then M G also. (2) If M is a 2-sum of M\ and M 2 , then M G ^rnax{2,u}- Proof. (1) Because 5i PI ^2 = 0, it follows that both Mj and M 2 consist of the union of components of M. Then, from (9) of Proposition 2.1.3 it follows that P{M) =max{/3(Mi),/1(M2)} < n. Hence M G also. (2) In this case |5'i nS' 2 | = 1. Let S 'in S '2 = {2 }. Then 2 is not a loop or a coloop of M l or M 2 . Also, since |5il < jSiA5'2| > IS2 I, it follows that jSij > 3 < IS 2 I. Our proof essentially consists of constructing a branch decomposition of M, of w idth < 71, from two optimal branch decompositions of Mi and M 2 , respectively. Let be an optimal branch decomposition of M{ for z = 1,2. Let u,- be the leaf of T{ th at is labelled by 2 and pi be the pendant edge incident with that leaf. Also let be the other vertex of T{ that is incident with p{. Since [S'il > 3 < l ^ l , it follows th a t both zui and 7V2 are trivalent.We form a ternary 107 tree T by joining the vertices w\ and u>2 with a new edge q and deleting the leaves nj and vo along with their pendant edges p\ and pg. We define a bijection I : L[T) — )■ S by letting (5-5) l{u) = /{(u), if a 6 L{Ti), for i — 1,2. Then we claim (T, I) is a branch decomposition of M. It suffices to prove that < n. Let e G E{T). First, suppose e = q. Then, from (4) and (5) of Lemma 5.5.1, it follows that p{Si\z) = pj{Si\z) = pi{Mi), where 1 < i < 2 and p{M) = Pi (M l) + P'zi^o) — 1, respectively. Therefore, rp{M, T, /, e) = p(Si\z) + p(S2 \z) - p{M) 4- 1 (5-6) = pi(Mi) + P2 {M2 ) - (pi(Mi) + P2 {M 2 ) - 1) + 1 = 2 < n. Next, suppose e / ç. Then, without loss of generality we may let e G £(Ti)\{pi}. Now one of the end-trees of e contains q as an edge and the other one does not. So, without loss of generality we may let Ae{TJ) — Ae{T\J\) and Be{T,l) = Be(Ti,li)AS2 . By (4) and (5) of Lemma 5.5.1, it follows that p{Ae{T,l)) — pi(Ae(Ti,/i)) and p(5e(T,/)) = Pi(Bc(Ti,/i))-f p 2 (-^ 2 ) “ 1, respectively. There fore, V,(M, T, Z, e) = PI(/le(Ti, Zi)) -b PI(Be(Ti, Zi)) -b P2(M2) - 1 - {Pli^l) + P2{M2) - 1) -b 1 ^ ^ - pi(/lc(Ti,Zi)) -bpi(J9e(Ti,Zi)) - pi(Mi) -b 1 = ^(Mi,Ti,Zi,e) < n. B The next 2 results are from Seymour [5]. Proposition 5.5.2. If M is a binary matroid and (% i, % 2 ) ^ 1 -separation of M then M is the 1-sum of M x % i and M x X 2 ; and conversely, if a binary matroid M is the 1-sum of M\ and M 2 then (S\,S2 ) is a 1 -separatioJi of M , and M\ and M 2 are proper minors of M . B 108 Proposition 5.5.3. If M is a binary matroid and (%%,%2) w an exact 2- separation of M then there are matroids M[, M 2 on X \ Uz, X 2 U z, respectively (where z is a new element), such that M is the 2 -sum of M\ and Mo. Conversely, if M is the 2 -sum of M\ and M 2 then {S\\S2 , S2 \S \) is an exact 2-separation of M, and M\, M 2 are isomorphic to proper minors of M . B Using the first 3 propositions, we prove a 3-connectedness result about a class of matroids that is defined by branch-width. Proposition 5.5.4. Suppose M is an obstacle to where n > 2. Then M is 3-connected. Proof. Assume that M is not 3-connected. Then M has a fc-separation, where 1 < k < 2. Therefore, either M has a 1-separation or M has an exact 2 - separation. Then, by Proposition 5.5.2 or Proposition 5.5.3, it follows that M is a 1 -sum or 2 -sum, respectively, of Mi and M 2 , which are its proper minors. Since M is an obstacle to it follows that P{M{) < n, for 1 < i < 2. There fore, by Propo.sition 5.5.1, it follows that P{M) < max{2, ra} = n, which is a contradiction. B 5.6. A reduction to 3-connected graphs and matroids Suppose M is a matroid such that /9(M) = 3. If M is not 3-connected then from the results of the previous section it follows that M is a 1- or 2-sum of two of its proper minors. At least one of these minors has branch-width 3. Hence, by induction, M has a 3-connected minor which has branch-width 3. Since 1 - and 2 -sums are to be included among the summing operations on ^ 3 it follows that in developing a theory for the finite generation of the lower ideal ^ 3 , we may without loss of generality assume that the binary matroids that have to be generated are all 3-connected. Similarly, in developing a theory for the finite generation of the lower ideal %, we may without loss of generality assume that the graphs that have to be generated are all 3-connected, because 0-, 1- and 2 -sums will be included among the summing operations on %. C H A P T E R V I A Structure Theory for 3-connected M atroids of Branch-width 3 6.1. Introduction In this chapter we develop a structure theory for 3-connected binary matroids and graphs that have branch-width 3. The symbols M and G always represent a binary matroid and a graph, respectively. In our arguments, we shall often use elementary results from matroid theory without exphcit mention. It is known that a circuit and a cocircuit of M m ust intersect in a set of even cardinality. Also, if A Ç S{M) then every circuit of M • A is also a circuit of M, and every circuit of M* • A is also a cocircuit of M. From these facts we deduce that every circuit of M • A must intersect every circuit of M* - A in a set of even cardinality. We shall often use this fact to obtain contradictions, and thereby prove our results. We also use the fact that if M is 3-connected then M is simple. Therefore, if A Ç 5(M) then |A| < - 1 and p{A) > [loggdAj 1 )]. 6.2. Small structures We begin with some fundamental definitions. Definition. Suppose F is a 4-element subset of M. F is S c i i d to be a wye-delta of M if F contains a triangle of M as well as a co-triangle of M. The subset F is said to be a A-circuit/bond of M if F is a 4-circuit of M as well as a 4-cocircuit of M. 109 110 We state some obvious facts. If D and E are a triangle and a cotriangle of M, respectively, that intersect each other, then D U E is a wye-delta of M. Also, if F is a wye-delta or a 4-circuit/bond of M then F is also a wye-delta or a 4-circuit/bond, respectively, of M*. Definition. Suppose A Ç S{M). Then (a) If |A| < 2 then A is said to be a trivial structure of M ; (b) Suppose )A| = 3. If A is a triangle or a cotriangle of M then A is said to be a 3-structure of M ; (c) Suppose |A| =4. If A is a wye-delta or a 4-circuit/bond of M then A is said to be a ^-structure of M ; (d) Suppose |A| = 6 . If M ■ A = M (/\ 2 ,3 ) and M* ■ A = M(C g) {triad-view), or M • A = M{C-i) © and M* ■ A = M (A '2 ,3 ) {triangle-view) then A is said to be a Q-structure of M. If A is a 3-, 4- or 6 -structure ofM then A is called a small structure of M . Both the rank and the corank of a 4-structure are equal to 3, and both the rank and the corank of a 6 -structure are equal to 4. Suppose A is a trivial or small structure ofM. Then, since |A| < 6 it follows that both M ■ A and M* ■ A are graphic matroids. The set A is also a trivial structure or a small structure, re spectively, of M*. Also, since M{C^)®M{C^) contains no 4-circuit and M{K 2 ,:i) contains no triangle it follows that no 6 -structure contains a 4-structure. Figure 6 . 1 illustrates the graphs that corresponds to the small structures. We prove a necessary and sufficient condition for a set to be a 4-structure. Proposition 6 .2 . 1 . Suppose M is 3-con,nected and A is a A-element subset of M. Then A is a 4-structure of M if and only if A contains a circuit as well as a cocircuit of M. Proof. If A is a 4-structure of M then, by the definition of a 4-structure, A contains a circuit as well as a cocircuit ofM. Conversely, suppose A contains a circuit D and a cocircuit E of M. Because M is 3-connected both D and E have I l l 3-structures 4-structures 6-structures Figure 6.1. The graphs corresponding to the small structures at least 3 elements each. The result now follows by the fact that M is a binary m atroid. B The next result provides us with a useful equation. Proposition 6.2.2. Suppose {A,B) is a separation of order 3 of M . Then (6-1) p{A) + p*{A) = \A\-\r2. Proof. This follows directly from the definition of ^{M ,A ,B ) given in Section 1.7. fl We prove necessary and sufficient conditions for 3- and 4-structures, in terms of the orders of the separations induced by them. Proposition 6.2.3. Suppose M is 3-connected and A Ç S{M ). Then (a) Let \A\ = 3. Then f^{A,B) = 3 if and only if A is a 3-structure. (b) Let |A| = 4. Then ^{A,B) = 3 if and only if A is a ^-structure. Proof, (a) From Proposition 6.2.2, it follows that p{A) -f p*{A) = 5. Since M is 3-connected, either p{A) = 2 and p*{A) = 3, or p{A) = 3 and p*{A) = 2. If the former holds then A is a triangle, and if the latter holds then A is a cotriangle, (b) From Proposition 6.2.2, it follows that p{A) -f- p*{A) = 6 . Since M is 3- connected, p{A) == 3 — P*{A). Therefore, A contains a circuit as well as a cocircuit of M. The result now follows by Proposition 6.2.1. fl 112 We conclude this section by writing down the canonical submatrices that cor respond to each small structure in M, where M is 3-connected. Given a small structureA of M it is possible to obtain the canonical submatrix for A by per forming elementary row operations and column permutations on a binary matrix that represents M. The first |A| columns of each matrix that we write down here will correspond to the canonical submatrix for A. The two matrices D\ and D 2 giving the canonical submatrices for a cotriangle and a triangle, respectively, are given by (6-2). In D\, only the first three entries of the third row are non-zero, indicating a co circuit of size 3. In Do, the sum of the first 3 column vectors is equal to the zero vector, indicating a circuit of size 3. 10 1 ... rl 0 1 - Oil... 0 1 1 1 1 1 0 0 0 0 (& 2 ) 0 0 0 ... D ‘2 = .0 0 0 0 Ô ... 0 The two matrices F\ and Fo giving the canonical submatrices for a wye-delta and a 4-circuit/bond, respectively, are displayed in (6-3). In F\ the sum of the first, second and fourth column vectors is equal to the zero vector, indicating a triangle. Also, in F\, only the first three entries of the third row are non-zero, indicating a cotriangle. In Tg the sum of the first four column vectors is equal to the zero vector, indicating a 4-circuit. Also, in Fi, only the first four entries of the third row are non-zero, indicating a 4-cocircuit. ((T3) rl 0 0 1 ...... - -1 0 0 1 ... . - 0 1 0 1 ...... 0 1 0 1 ... . 1 1 1 0 0 ... 0 1 1 1 1 0 . ' 0 0 0 0 0 ...... F2 = 0 0 0 0 ... . .0 6 0 0 ...... 6 0 Ô 0 ... . The matrix X\ giving the canonical submatrix for a 6-structure in triad view is displayed in (6-4). In this matrix only the first three entries of the third row 113 are non-zero, indicating a cotriangle. In the fourth row, only the fourth, fifth and sixth entries are non-zero, indicating another cotriangle that is disjoint from the first one. The first, second, fourth and fifth column vectors represent a 4-circuit of M, as do the second, third, fifth and sixth column vectors, and the first, third, fourth and sixth column vectors. rl 0 0 1 0 0 ■ 0 1 0 0 1 0 1 1 1 0 0 0 Ô ! ’ 0 (6-4) Xi 0 0 0 1 1 1 0 . . 0 0 0 0 0 0 0 .0 0 Ô 0 0 0 The matrix X 2 giving the canonical submatrix for a 6-structure in triangle view is displayed in (6-5). In this matrix the sum of the first three column vectors is equal to the zero vector, indicating a triangle. Another triangle, disjoint from the first one, is indicated by the fact that the sum of the fourth, fifth and sixth column vectors is also equal to the zero vector. An examination of the tliird and fourth rows shows the existence of two non-disjoint 4-cocircuits. By adding these two rows we observe the existence of the third 4-cocircuit, which is also not disjoint with the other two 4-cocircuits. rO 0 0 1 0 1 0 0 0 0 1 1 1 0 1 1 0 1 Ô ' . Ô 1 (&.S) X 2 0 1 1 0 1 0 . . 0 0 0 0 0 0 0 .0 Ô 6 Ô ÔÔ 6.3. Complete 3-separability of sets, of size < 5 We estabhsh 2 results on complete 3-separabihty. Proposition 6.3.1. Suppose A Ç S{M ). Then, if A is a trivial structure, 3- structure or 4-structure, A is completely 3-separable in M . Proof. This follows from Propositions 2.2.1 and 6.2.3. ■ 114 Proposition 6.3.2. Suppose M is 3-connected and A is a b-elem.ent subset of M such that = 3. Then A contains a A-structure of M , and A is completely 3-separahle in M . Proof. From Proposition 6.2.2, it follows that p{A) + p*{A) = 7. Also, from the 3-connectivity of M , it follows that either p{A) = 3 and p*{A) = 4, or p{A) = 4 and p*{A) = 3. Without loss of generahty, we assume the former. Then M ■ A is isomorphic to the polygon matroid of the graph that is obtained by joining two of the non-adjacent vertices of a 4-circuit. Thus M ■ A contains a unique 4-circuit. Looking at M* • A, we first observe that it cannot be a 5-circuit. If M* ■ A contains a triangle then that triangle must intersect a triangle of M • A, and then A contains a wye-delta of M. Otherwise, M* ■ A consists of a 4-circuit and a coloop. If this 4-circuit has the same set of elements as the 4-circuit oi Ad • A then A contains a 4-circuit/bond. Otherwise, this 4-circuit of M • A contains a triangle of Ad* ■ A, which is a contradiction. The complete 3-separabihty of A follows from Proposition 6.3.1. ■ 6.4. Six structures We begin by stating a graph theoretic result for 6-element simple binary ma troids (without proof). Proposition 6.4.1. Suppose Ad is a simple binary matroid that consists of 6 elements. Then (1) If p{Ad) = 3 then Ad ^ Ad{IU); (2) If p{Ad) = 4 then Ad is isomorphic to one of the following: /o/ M (C 3)@M(C 3); M(A:2,3); (c) the direct sum of U\^\ with the polygon matroid of the graph that is formed by joining two non-adjacent vertices of a 4- circuit; 115 (d) the polygon matroid of the graph that is formed by joining two non-adjacent vertices of a 5-circuit. {See Figure 6.2). (3) If p{M) — 5 then M is isomorphic to the direct sum of with the polygon matroid of a t-circuit where i = 3,4,5 or 6. Proposition 6.4.2. Suppose M is 3-connected and A Q S{M ) such that |A| = 6 and ^(A, B) — 3. Then either A contains a 4-structure of M or A is a Q-structure of M , but not both. Proof. That A cannot both he a 6-structure and contain a 4-structureof M , follows from our remarks which appeared after the definition of trivial and small structures. From Proposition 6.2.2, it follows that p(A) -t- p*(A) = 8. Because M is 3- connected, the only possibihties are p{A) — 3 and p*{A) = 5, or p{A) = p*{A) — 4, or p{A) = 5 and p*{A) = 3. Without loss of generality, it suffices to consider only the first two. Case (i): p{A) = 3 and p*{A) = 5. From (1) of Proposition 6.4.1, it follows that M • A ^ M{K^)., and from (3) of that result M* • A contains a circuit that consists of 3, 4, 5 or 6 elements, with all the other elements being coloops. Now, 2 out of any 3 elements of M • A are contained in a triangle of M. Therefore, if M* • A contains a triangle then A contains a wye-delta of M. Next, suppose M* • A consists of a 4-circuit and 2 coloops. Then the 2 elements of M • A which correspond to the coloops of M* • A cannot be contained in a triangle of M • A. This is because, if that happens then the third element of that triangle will be the only element of that triangle that is contained in the 4-circuit of M* ■ A. Hence the 4-circuit of M* ■ A must also be a 4-circuit of M ■ A. Therefore, A contains a 4-circuit/bond of M. As for the other 2 possibihties for M* ■ A, we see that they cannot happen, because any 5 elements of A contains a triangle of M. Case (ii): p{A) — p*{A) = 4. Figure 6.2 illustrates the graphs that correspond to the possible matroids that are given by (2) of Proposition 6.4.1. We denote these graphs by G gj, Gg g, 116 and Gg^ 4, respectively, as shown in the figure. We also let for 1< 2 < 4. Gs,! Ge,2 Ge,3 F ig u re 6,2. The graphs such that = M[G%^i), for 1 < i < 4 Due to the symmetry between M ■ A and M* ■ A, it suffices to consider the cases that arise when M ■ A = and M* ■ A = M(;where 1 < f < 4 and f < A; < 4. First, suppose M ■ A = Mgj. Then M* ■ A cannot contain a triangle because that triangle must intersect one of the 2 disjoint triangles of M A in an odd number of elements. Therefore, the only possibility is M* ■ A = Mg 2- T hen A is a 6-structure of M. Next, suppose M ■ A = Mg^ 2- If M* • A = Mg ^2 or M* • A = Mg^g, let F be one of the 4-circuits of M* ■ A. Then, since F must intersect each of the three 4-circuits of M A in a set of even cardinahty, F must correspond to a 4-circuit of M • A. Hence A contains a 4-circuit/bond of M. We conclude this possibihty for M • A by showing that M* ■ A % Mg 4. Assum e M* • A = Mg 4. Let P and Q denote the 5-circuit of M* ■ A and a 4-circuit of M • A, respectively. Then | f U Q| + |P H Q| = |f | -f- |Q| = 9 and 5 < |P U <51 < 6. If |P U Q| =6 then |P fl Q| = 3, which is a contradiction. If |P U Q| = 5 then Q Q P. Hence P must intersect one of the other two 4-circuits of M A in a set of 3 elements. This is a contradiction. Next, suppose M ■ A = Mg_g. If M* ■ A = Mg^g then every triangle of M • A must intersect every triangle of M • A. Hence A contains a wye-delta of M. To conclude tliis possibility for M-A, suppose M* - A = Mg^ 4. Let D be a triangle of M • A. T hen D must intersect the triangle of M* ■ A, because otherwise D would 117 be contained in the 5-circuit of M* ■ A, and that is a contradiction. Therefore A contains a wye-delta of M. Finally, suppose M ■ A = Mg ^4 = M* ■ A. Now M ■ A has a triangle D, as in the case just discussed. Therefore, by the preceding arguments, it follows that A contains a wye-delta of M. B We are now able to prove a result about complete 3-separabihty. Proposition 6.4.3. Suppose M is 3-connected and A Ç S{M) such that (>1| < 6. Suppose also that if |A| > 3 then £,{A,B) — 3. Then A is completely 3-separable in M. Proof. By Propositions 6.3.1 and 6.3.2, it suffices to prove this only for |A| = 6. Also, by Proposition 6.4.2, it is enough to prove this when A is a 6-structure. Since A is then the union of 2 disjoint triangles or cotriangles, and each of these is completely 3-separable, it follows that A is also completely 3-separable. B We estabhsh a necessary condition for a 3-connected binary matroid to have branch-w idth 3. Proposition 6.4.4. If M is 3-connected and (i{M) = 3 then M has a A-structure or a Q-structure. Proof. Since (5{M) = 3 it follows that M has at least 6 elements. Then, by the ht tie twig lemma (Proposition 1.3.7) it follows that 3A Ç S{M) such that 4 < |A| < 6 and f{A,B) = 3. Now from Propositions 6.2.3, 6.3.2 and 6.4.2 the result foUows. H 6.5. Applications to small binary matroids In this section we apply the theory of small structures to binary matroids that have no more than 10 elements. 118 Proposition 6.5.1. If M has at most 9 elements then f}{M) < 3. Proof. From (2) of Proposition 2.3.1, it follows that the result is true if |5(M )| < 7 or m in{p{M ), p*{M)} < 3. Therefore, we may suppose that 8 < |5(M)|. Also, without loss of generality let p{M) — 4. If M is not 3-connected then, by Propositions 5.5.2 and 5.5.3, it follows that M is a 1 -sum or a 2-sum of 2 of its proper minors. Therefore, by Proposition 5.5.1, it suffices to prove the result for 3-connected M. First, suppose M has a triangle, say A. Then, from Proposition 6.3.1, it follows th at A is completely 3-separable. Also, since ^{A,B) = 3 and \B\ < 6, from Proposition 6.4.3 it follows that B is completely 3-separable. Hence (i{M) — 3. Next, suppose M has no triangle. Then any 4-element basis of M may span only 8 elements. Therefore, \S{M)\ — 8 and a standard matrix representation of M with respect to a basis is given by 1 0 0 0 0 1 1 1-1 0 1 0 0 1 0 1 1 (6-6) 0 0 10 110 1 0 0 0 1 1 1 1 OJ Now M contains a 4-circuit/bond, say A (the elements of M corresponding to the 4th, 5th, 6th and 7th columns of this representation, for instance). Then since \A^\ = 4, by (b) of Proposition 6.2.3, it follows that A^ is also a 4-circuit/bond. Therefore, by Proposition 6.3.1, it follows that /3{M) = 3. B Proposition 6.5.2. Suppose M is 3-connected and consists of 10 elements. Then, M has a 4-structure if and only if M is not an obstacle to branch-width 3. Proof. First, suppose M has a 4-structure, say A. Then, by Proposition 6.4.3, both A and A^ are completely 3-separable. Therefore (3{M) = 3 and M is not an obstacle to branch-width 3. Next, suppose M is not an obstacle to branch-width 3. Then, by Proposition 6.5.1, it follows that (3{M) — 3. By the central-edge-lemina (Proposition 1.3.5), there is a separation (A,B) of M such that (,{A,B) = 3 and |A| > 4 < |H|. If A| = 4 or |H| = 4 then, by (b) of Proposition 6.2.3, it follows that M has a 119 4-structure. Also, if |A| — \B\ = 5 then, by Proposition 6.3.2, it follows that M has a 4-structure. B 6 . 6 . An upper bound for |5'(M')|, for simple M E ^ 3 Suppose M is simple and (3{M) = 3. Then, by Proposition 2.3.2, it follows that |S(M)| < 3 \/ 2 '’(^^)+l , where r{M) = min{p(M),p*(M)}. In tliis section we obtain a hnear upper bound for |5(M )|, in terms of r[M). We begin with a useful lemma. Lemma 6.6.1. Suppose A Ç S(M ) and t E Suppose also that A is the union of t distinct cocircuits of M , none of which is contained in the union of the others. Then p{A^) < p{M) — t. Proof. We prove this by induction. Since the complement of a cocircuit is a hyperplane, the result is true when t = 1. Assume true when t — p, where p is a positive integer, and suppose t = p-\-l. Let A = L>] U ■ • • U Dp U , where D i,..., Hp+i are cocircuits of M, that satisfy the hypothesis. Then — p{M) — {p + 1). B We prove 2 results about |5(M )| when M is simple and has low branch-width. Proposition 6.6.1. If M is simple and /3(M) = 1 then |5(Af)| = p{M). Proof. Tliis follows from (4) of Proposition 2.1.3 and (2) of Proposition 2.1.5. B In our proofs during the remainder of this section, we let m = p{M) and if {A,B) is a separation of M then we let a = p{A) and b — p{B). Then a + b = m + I,{A, B) — 1. 120 Proposition 6.6.2. Suppose M is simple with p{M) > 1 and /?(M) < 2. Then |5'(M)| < - 1 . Proof. If (3{M) = G then |5'(M)| = 1 and the result is true. If (3{M) = 1 then the result foUows, by Proposition 6.6.1. So suppose j3{M) = 2. Then, by (2) of Proposition 2.3.1, it follows that p{M) > 2. We proceed by induction on r = p{M). Since the result is true for p{M) = 2, we assume true for p(M) < r, where r > 3. First, suppose M is disconnected. Then M has a 1-separation {A,B). There fore, a+b — m and a < r —1 > b. Then, by the induction hypothesis, |A| < 2a — 1 and \B\ <26 — 1. Therefore, |S'(M)| = |A| + \B\ < 2(a -f 6 ) — 2 < 2m — 1. Next, suppose M is connected. Then M is the polygon matroid of a simple series-parallel network G, where |F(G)| = m-f 1. Now M can be constructed from a simple series-parallel network G\, where |y(Gi)| = m, either by subdividing an edge of G\, or by adding an edge parallel to an edge of G\ and subdividing that edge. By the induction hypothesis, |J5(Gi)| < 2(m — 1) — 1 . By the above described construction, |5(M )| = \E{G)\ < |F7(Gi)| -f 2 = 2 ?n — 1 , as required. ■ The next result turns out to be a special case of the main result. Proposition 6.6.3. Suppose M is simple, with p[M) — 4 and P{M) = 3. Then |S (M )| < 1 1 . Proof. If M is disconnected then M has a 1-separation {A,B). Then a -b 6 = 4 and a > 1 < 6 . Therefore, without loss of generality, a = 1 and 6 = 3, or a = b = 2. Hence |5(M)| = \A\ -f |H| < max{l -(- 7,3 -f- 3} = 8 . If M is connected but not 3-connected then M has a 2-separation (A, B). Then a + 6 = 5 and a > 2 < b. Therefore, without loss of generality, a = 2 and 6 = 3. Hence \S{M)\ = |A |-b |B | < 3 + 7 = 10. Finally, suppose M is 3-connected. If M has a 6 -structure, say A, then since A is the union of 2 distinct cocircuits, by Lemma 6.6.1, it follows that p{A^) < 2. Therefore, \S{M)\ < 6+3 = 9. On the other hand, suppose M has no 6 -structure. We may, without loss of generahty, suppose |5'(M)| > 5. Then, by the ht tie twig 121 lemma and Proposition 6.3.2, it follows that M has a 4-structure, say A. Since A contains a cocircuit, by Lemma 6.6.1, it follows that p{A^) < 3. Therefore, |5 (M )| < 4 + 7 = 11. H Proposition 6 .6 .4 . Suppose M is simple with p{M) > 2 and (3{M) < 3. Then |S'(M)| < 4p(M) - 5. Proof. If p{M) > 2 then p{M) < 2p{M) — 1 < 4/)(M) — 5. Therefore, by Proposition 6.6.2, it follows that the result is valid if P{M) < 3. Suppose (3{M) — 3. The result is tru e for p{M) — 2 because 2^ — 1 = 4.2 — 5, it is true for p{M) — 3 because 2^ — 1 = 4.3 — 5, and it is valid for p{M) — 4, by Proposition 6.6.3. We proceed by induction on r = p(M), assuming the result to be true for p{M) < r where 7- > 5. If M is disconnected then M has a 1-separation (A,B). T hen a b = m and m — l> a > l< 6 < 77i — 1 . If a = 1 then |A| = 1 and 6 > 4. Since P{M ■ B) < 3, by the induction hypothesis, \B\ < 46 — 5 = 4m — 9. Therefore, |5(M)| = 1 + |S| < 4m — 5. If a > 1 then, since P{M • A) < 3 > (3{M ■ B ), by the induction hypothesis |A| < 4a — 5 and \B\ <46 — 5. Therefore, \S{M)\ < 4{a + 6 ) — 10 < 4m — 5. Next, suppose M is connected but not 3-connected. Then M has a 2 -separation (A, B). Therefore, a + 6 = m + 1 and m — 2>a>2<6 Finally, suppose M is 3-connected. Then by Proposition 6.4.4, it follows that 3X Ç 3{M) such that X is either a 4-structure or a 6 -structures ofM. Let Y = % + First, suppose X is a 4-structure. Then |X| = 4 and, by Lemma 6.6.1, it follows that p{Y) < r — 1 . Since M ■ Y is simple, by the induction hypothesis and Proposition 6.6.2, it follows that |y| < 4(r — 1) — 5. Therefore, \S{M)\ — 4 + \Y\ < 4 r - 5 . Next, suppose X is a 6 -structure. Then |X| = 6 and, by Lemma 6.6.1, it follows that p{Y) < r — 2. Since M -Y is simple, by the induction hypothesis 122 and Proposition 6 .6 .2 , it follows that |y| < 4 (7’ — 2 ) — 5. Therefore, |5(M )| = 6 + |F | < 4r - 7. a Proposition 6.6.5. üet r(M) = min{p(M),p*(M)}. Suppose M is ^-connected with r{M) > 2 and p{M) ^ 3. Then \S(M)\ < 4r(M ) - 5. Proof. Since M is 3-connected both M and M* are simple. Also, by (2) of Proposition 2.1.3, it follows that P{M*) — 3 also. The result now follows from Proposition 6.6.4. a 6.7. Applications to 3-connected graphs It is easy to see that wye-deltas occur frequently in graphic matroids. It is also common to observe 6 -structures isomorphic to M(/V9 ,3 ) as submatroids of graphic matroids. In this section we show that there is only one graph G such th a t M[G) has a 4-circuit/bond, and only 2 graphs G such that M{G) has a 6 -structure that is isomorpliic to M{G^) ® M{C2,)- Using these facts we prove a result about 3-connected graphs on 12 edges that is useful later. Proposition 6.7.1. Suppose G is “i-connected. Then (1) The matroid M{G) has a 4-circuit/bond if and only if G = / I 4 ; (2) The m,atroid M{G) has a Q-structure A such that M ■ A ^ M{C‘/)®M{C/) if and only if G = W 4 or G is isomorphic to the 3-prism. {See Figure 6.3). IÛ 3-prism Figure 6.3. The graphs A 4 , W4 and the 3-prism 123 Proof. (1) If G = J\ 4 then M{G) has a 4-circuit/bond because any 4-circuit of G is also a bond of G. Conversely, suppose M{G) has a 4-circuit/bond, say A. Let H = G ■ A. Then, since G is 3-connected, each of the 4 vertices of H must be a vertex of attachment of H. Therefore, ^{M{G), E{H), E{H)) = 3 but f,'(G,H,H) = 4. Then, by Lemma 4.4.1, it follows that \E{H)\ = 2. Therefore G = JQ. (2) We first show that both W 4 and the 3-Prism have 6 -structures in triangle view. Let G = IL4 and iJ be a subgraph of G that consists of any two 3-circuits that have exactly one vertex in common. Also, let A = E[H). Then, since p{A) = 4, p{B) = 2 and p{M[G)) = 4, it foUows th at f,{M(G),A,B) = 3. Therefore, since A contains no 4-structure of M (G ), it foUows th at A is a 6 -structure of M. Also, M(G)A = M(G3)eM(G3). Let G be isomorphic to the 3-Prism and H be the subgraph of G that consists of the two 3-circuits. Also, let A = E{H). Then, since p{A) = 4, p{B) = 3 and p{M{G)) = 5, it foUows th a t ^{M{G), A, B) = 3. Therefore, since A contains no 4-structure ofM{G) it foUows that A is a 6 -structure ofM. Also, M{G) • A = M(G3)@M(G3). Conversely, suppose M{G) has a 6 -structure A such that M{G) ■ A = M{C‘^) 0 M{C^). Let H — M{G) ■ A. T hen H consists of two 3-circuits. Because G is 3-connected, these two 3-circuits may have at most one vertex in common. Also, the other vertices of these two 3-circuits must be vertices of attachment of H in G. If the two 3-circuits have exactly one vertex in common then H has 4 vertices of attachment. Then, by Lemma 4.4.1, it foUows that G = W4 . If the two 3-circuits have no vertex in common then LT has 6 vertices of attachment. Then, by Lemma 4.4.1, it foUows that G is isomorphic to the 3-prism. B Definition. If A is a trivial or small structure of M(G) then we shaU refer to the subgraph G A of G, as the corresponding trivial or small structure of G. Proposition 6.7.2. (1) If G has at most 9 edges then (i'[G) < 3; 124 (2) Suppose G has exactly 10 edges and is 3-coimected. Then G is not an obstacle to graphic branch-width 3 if and only if G has a wye-delta; Proof. (1) By our results in Chapter 5 we may, without loss of generality, assume th a t G is 3-connected. Then the result follows by Propositions 4.4.1 and 6.5.1. (2) The result follows by Propositions 4.4.1, 6.5.1, 6.5.2 and 6.7.1. ■ Proposition 6.7.3. Suppose G has 12 edges, is 3-connected and has branch- width 3. Then, if G has no wye-delta then G = /^ 4 ,3 . Proof. By (1) of Proposition 6.7.1, it follows that M{G) has no 4-structure. Therefore, by the central-edge-lemma and Propositions 6.2.3 and 6.3.2, it follows th a t M{G) has no separation {A,B) of order 3 such that 4 < |A| < 5 or 4 < \B\ < 5. Hence M{G) has a separation {A,B) of order 3 such that |A| = 6 = \B\. Therefore, by Proposition 6.4.2, both A and B are 6 -structures ofM{G). Also, by (2) of Proposition 6.7.1, it follows that A % M(Cg) @ M(Cg) g B. Therefore, A = M (A '2 ,3 ) =B. Let ff = G -a and K = G- B. Then H ^ ^ 2 ,3 = K. Also, from Lemma 4.4.1, it follows that [H,K) is a 3-separation of G. Now, from the 3-connectivity of G it foUows th at G = Ka,2 - ■ Proposition 6.7.4. ( 1 ) If G has at most 9 edges then P’{G) < 3; (2) Suppose G has exactly 10 edges and is 3-connected. Then G is not an obstacle to graphic branch-width 3 if and only if G has a wye-delta; (3) Suppose G has exactly 12 edges and is 3-connected. Siippose also that G % A"4^3 and G has no wye-delta. Then P'{G) > 3. Proof. ( 1 ) By our results in Chapter 5, we may, without loss of generality, assume th a t G is 3-connected. Then the result foUows, by Propositions 4.4.1 and 6.5.1. (2 ) The result foUows, by Propositions 4.4.1, 6.5.1, 6.5.2 and 6.7.1. (3) The result foUows, by Proposition 6.7.3. ■ Suppose G is 3-connected. If H is a wye-delta of G then H has one mono valent vertex, two divalent vertices and one trivalent vertex. The vertex that is monovalent in H has valency > 2 in H. The trivalent vertex of H is the vertex 125 that is not a vertex of attachment oi H . If ff is a 6 -structure ofG in triad view, then the three independent vertices of H are the vertices of attachment of H. (See Figure 6.4). -f W ye-delta 6-structure Figure 6.4. A wye-delta and a 6 -structure in G Proposition 6.7.5. The cube, the octahedron, and Vg; each has graphic branch-width > 3. (5ee Figure 6.5). Also, is an obstacle to graphic branch- width 3. cube octahedron Figure 6.5. The graphs of the polygon matroid obstacles to branch-width 3 Proof. This follows from (2) and (3) of Proposition 6.7.4. ■ Remark. In chapter 7 we prove that all 4 graphs mentioned in Proposition 6.7.5 are obstacles to graphic branch-width 3. Figure 6 . 6 consists of a drawing of and a geometric representation of M{K^). The vertices of are denoted by the integers from 1 to 5. The points of the geometric representation are denoted by pairs of the form ij, where the pair ij corresponds to the edge of that joins the vertices i and j . 126 2 (a) A'; 12 23 (b) M(A, F igu re 6 .6 . /1 5 and M { K ^ ) CHAPTER VII The Lower Ideal % 7.1. Introduction 0-, 1-, 2- and wye-delta 3-sums Figure 7.1. The lower ideal % From Proposition 5.4.7 it follows that % = The goal of this chapter is to establish the structure of as claimed by Figure 7.1. However, as explained in Section 5.6, we shall only be concerned with 3-connected graphs. Also, the only summing operation of interest will be the wye-delta 3-sum, which is defined in Section 7.3. The internal structure of % is dealt with in the next seven sections, while the external structure is estabhshed in the last section. In the next section, we 127 128 prove a condition for a triad or a triangle (of a graph of branch-width 3) to be localizable. This result is then utilized in Section 7.3 to prove that the wye-delta 3-sum is closed in The wye-delta 3-sum generators are estabhshed in Section 7.4. We also show that the 5-wheel is not wye-delta 3-sum decomposable, although it satisfies cer tain necessary conditions for decomposabihty. In this section we prove that the unique non-generator on 10 edges is decomposable. We also introduce the con cept of a big 3-separation, which is a 3-separation in which both subgraphs have at least 5 edges, and prove that every non-generator with at least 11 edges has such a big 3-separation. The next 4 sections are devoted to proving that all non-generators are wye- delta 3-sum decomposable. We use the fact that G is wye-delta 3-surn decom posable if and only if it has a big 3-separation [H,K) and proper 3-connected wye-delta H- and K- extension minors. In Section 7.5 we define robust separa tions. These are separations whose subgraphs may be “extended” to 3-connected minors by “tagging on” wye-deltas at the vertices of attachment of these sub graphs. In Section 7.6 we introduce push operations which enable us to obtain robust big 3-separations from non-robust ones. In Section 7.7 we prove that all wheels with at least 6 spokes are decomposable. The proof that all non-generators are decomposable is given in Section 7.8. The symbol G, with or without suffixes, shall always denote a graph. As explained earher graphs are 3-connected, almost without exception. Due to this, G is simple, connected and has no cut-vertices or 2-separations. Also, li v e V{G) then valg(u) > 3. We shall use many such consequences of 3-connectivity in order to prove our results without exphcitly mentioning the 3-connectivity of G. 7.2. The localizability of triads and triangles of graphs We begin by defining the concept of localizability for matroids and graphs. Definition. Suppose W Ç S{M). T hen W is said to be (branch-width) localiz able for M if 3 an optimal branch decomposition (T,/) of M and e G E{T) such that either Ae = W or Be = W. Also, (T, I) is called a localization of W. 129 Definition. Suppose H is a subgraph of G. Then H is said to be (graphic branch-width) localizable for G if 3 a graphic optimal branch decomposition (T, I) of G and e G E{T) such that either Ag = E{H) or Be = E{H). Also, (T ,/) is called a localization of H. We define two graphs that are significant in the theory of localizabihty of 3-connected graphs of branch-width 3. Definition. The simple graph G, known as the octacube^ consists of 7 vertices and 12 edges. The vertex-set of G consists of 3 pairwise disjoint subsets Vj, V2 and V 3 such that jVi| = = 3 and |Vj| = 1. Every vertex of Vi is joined to every other vertex of Vj, forming a triangle. Every vertex of V2 is joined to two distinct vertices of V\ so that every vertex of Vj has valency 4. Finally, the single vertex of V3 is joined to all three vertices of V^- The triangle of G wliich has Vj as its vertex-set is called the distinguished triangle of G. The triad of G which has the single vertex of V 3 as its trivalent vertex is called the distinguished triad of G. (See Figure 7.2). We note that the distinguished triad and the distinguished triangle of the octacube are the only triad and triangle, respectively, which are not subgraphs of any wye-delta of the octacube. Definition. The graph G formed by joining any two non-adjacent vertices u and n of a graph that is isomorphic to 7Q 3 , is called the augmented K^^^-graph. The triad of G, no edge of which is incident with either u or u, is called the distinguished triad of G. (See Figure 7.2). The distinguished triad of the augmented Ag 3 graph is the only triad which is not a subgraph of any wye-delta of that graph. We establish some more terminology. Definition. First, suppose is a triad of G. Then G is said to have a forbid den octacube minor (respectively, augmented Ag g minor) w ith H fixed, if the octacube (respectively, augmented A'g g graph) is a minor of G including H and w ith H corresponding to the distinguished triad of the octacube (respectively, augmented /fg g graph). 130 octacube augmented 1(3 ,3 graph Figure 7.2. Two significant graphs Next, suppose H is a, triangle of G. T hen G is said to have a forbidden octacube minor w ith H fixed, if the octacube is a minor of G including H and w ith H corresponding to the distinguished triangle of the octacube. Lemma 7.2.1. Suppose G is 3-connected, with P(G) = 3. Then ( 1 ) Suppose Y is a triad of G. If G has no forbidden octacube minor or for bidden augmented minor with Y fixed, then Y is localizable for G. (2 ) Suppose Z is a triangle of G. If G has no forbidden octacube minor with Z fixed, then Z is localizable for G. Proof. In our proof we shall use the fact that fI{G) = /3'(G), which follows from Proposition 4.4.1. Before we begin proving the two assertions of the lemma, we establish some notation and facts. The third edge of a triangle is spanned by the other 2 edges, and the trivalent vertex of a triad is incident only with the 3 edges of that triad. Due to these facts, if 2 of the 3 edges of a triangle or a triad can be localized then the triangle or triad, respectively, can also be localized. Therefore, without loss of generality, we assume that no 2 edges of the triangle or triad in question are locahzed. Let X\, X 2 and X 3 be the 3 edges of the triad or triangle of G which is to be locahzed. Suppose (T, I) is a graphic optimal branch decomposition of G. Let t be the unique trivalent vertex of T such that the 3 branches T\, Tg and Tg of T at t contain the T-leaves that are labelled by %%, X 2 and X 3 , respectively. Let be the subgraph of G such that the set of T-leaves of are labelled by E{Hi), for 131 1 < z < 3. (See Figure 7.3). Then X,- G E{Hi). Also, let H\ = G ■ {E{Hi)\Xi). Without loss of generality, let \E{Hi)\ > \E{H2)\ > \E{H2)\. Then, by our localization assumption, \E{H\)\ > \E{H2) \ > 2. Figure 7.3. The branches of T at t Let Wi = W[G,Hi)^ for 1 < i < 3. Then 2 < \Wi\ < 3, with |W)| = 2 if and only if \E[Hi)\ = 1. Let n = \Wi fl W2 I. Then n < 3. We show that n > 2, as follows. IW1 AW 2 I = \Wi\ + iWo] - \Wi n W 2 I = 6 - 2n. Since W 1 AW 2 C W 3 it follows that 6 — 2n < 3. Therefore, n > 2. In our proofs we consider n — 2 and n = 3 separately. From Proposition 6.4.3 it follows that all triangles and triads of G are localiz able if |F(G)| < 9. Hence, we assume that \E{G)\ > 10. (1) Let V{Y) — ( z i, Z2 , zg, )/}, with y being the trivalent vertex and the vertex X{ being incident with the edge for 1 < i < 3. Also, let H be the subgraph of G such that W{G,H) = {a;i, «2 i Eind H U Y = G. Then val//(zf) > 2. Also, y £ W \f] W2 n W3 . (See Figure 7.4). First, suppose n = 2. We show that \E{H2 )\ > 2. If \E{H2 )\ = 1 then E{H2 ) = {A 3 }. Then, since IW 1 AW 2 I = 2 and = 2, it follows that the incident vertices of A 3 are the vertices of W 1 AW 2 . Tliis is a contradiction because A 3 is incident with y ^ W\AW 2 - If |F (f/’3 )| = 2 then, without loss of generality, zg G V{H\). Let A 3 be the second edge of ffg. (See Figure 7.5(a)). We can then obtain a localization of Y after an intermediate graphic optimal branch decomposition. (See Figure 7.5(b) and (c)). 132 X2 Figure 7.4. The triad Y in the graph G © (b) (a) (c) Figure 7.5. Obtaining a localization of Y for a graph G If \E{H)\ > 3 then zg ^ V{H\) U V{H2 )- Suppose p, rj', r G V{G) such that W\ n W2 = n VFg = {y,p} and IVg n IF) = {y,q}. (See Figure 7.6). If |jB(I7g)| = 3 then zg is adjacent to both p and q. Let Q G E{G) such that E{H^) = {Xg,P, Q}, with P and Q being adjacent with p and q, respectively. (See Figure 7.7(a)). We obtain a localization of Y after an intermediate graphic optimal branch decomposition. (See Figure 7.7(b) and (c)). As a final subcase, suppose |F^(ffg)| > 4. Then H[ has a cycle C, that is 3- 133 Figure 7.6. The triad Y in the graph G Q P P Figure 7.7. Obtaining a localization of Y for a graph G joined to the vertices of attachment of H'-, for 1 < ?’ < 3. T hen G has a forbidden octacube minor with the triad Y fixed. (See Figure 7.8). Next, suppose n = 3. Then W\ — W2 D li/3 . If |JS (^ 3 )| = 1 then Tfg = G {%3 }. (See Figure 7.9(a)). We obtain a locahzation of F after an intermediate 134 ^2 Figure 7.8. The graph G with a forbidden octacube minor with Y fixed graphic optimal branch decomposition. (See Figure 7.9(b) and (c)). (b) y 2 (a) (c) Figure 7.9. The graph G, and obtaining a localization of Y If \E{H^)\ — 2 then H^) — 3. If is the edge of If3 that is distinct from % 3 then must have the two vertices of {W\ fl W2 )\y as its adjacent vertices. (See Figure 7.10(a)). We obtain a localization of F after an intermediate graphic optimal branch decomposition. (See Figure 7.10(b) and (c)). 135 H' XI y Figure 7.10. The graph G, and obtaining a localization of Y If \E{H-^)\ > 3 then 3u,v G V{G) such th at W\ = ITg = = {u, y,î/}. Then X{ ^ {%,%}, for 1 < i < 3, because otherwise \E{Hi)\ < 2, for th at i. We show th at G = if \E{G) = 9, and th at G lias a forbidden augmented lig g minor with Y fixed, otherwise. If \E{G) = 9 then \E{H\)\ — \E{H2)\ = \E{H^)\ — 3. Since each vertex of G has valency at least 3 it follows that |V(G)| < 6 . Also, since V{Hi) 3 {u, i), y, Xj} and the latter set is of cardinahty 4, it follows that \V{Hi)\ > 4, for 1 < i < 3. Hence |H(G)| = 6. Therefore, G = ATg g. On the other hand, if \E{G)\ > 10 then \E{Hi)\ > 4. Therefore, iJj has a cycle C that is 3-joined to xj,, u and v. From this it follows that G has a forbidden augmented fCg^g minor with Y fixed. (See Figure 7.11). (2) Suppose G has no forbidden octacube minor with Z fixed. Assume Z is not localizable for G. Then E{Z) is not localizable for M{G). Therefore, E{Z) is not localizable for [M{G))*. Since M(G) is a grapliic matroid, so are all its minors. Therefore, {M{G))* cannot have a forbidden augmented Afg^g minor with E{Z) fixed. Therefore, by (1), it follows that (M{G))* must have a forbidden octacube minor with E{Z) fixed. Hence (M(G)) must have a forbidden octacube minor 136 H'2 Figure 7.11. Graph G has forbidden augmented A'g g minor with Y fixed w ith E{Z) fixed. Therefore, G must have a forbidden octacube minor with Z fixed. This is a contradiction. ■ 7.3. W ye-delta 3-summing We begin with some prehminary definitions. Definition. Suppose (H,K) is a separation of G such that \E(H)\ > 2 and = 3. Let V{H n K) - {x.ij.z}. The graph G\ constructed from H by joining x, y and z to a new vertex w, and joining the vertices y and z is called a wye-delta H-extension at (x, {y, z}), or simply a wye-delta H-extension. If G\ is 3-connected then it is called a 3- connected wye-delta H-extension. (See figure 7.12). Suppose G has a minor where G' fixes LT, and G' is a wye-delta iî-extension (at (x,{y,z})). Then G is said to have a wye-delta H-extension minor (at (x,{y, z})). If G' is a proper minor or a 3-connected minor then it is called a proper wye-delta H-extension minor or a 3-connected wye-delta H-extension minor, respectively. In this case K' denotes the wye-delta subgraph of G' such 137 Figure 7.12. A wye-delta iï-extension G\ at {x,{y,z}) th a t W {K ', k ') = {z, ?/, z}. Similarly, for a wye-delta A”-extensioii miner, H' denotes the corresponding wye-delta subgraph of that minor. (See Figure 7.13). w H Figure 7.13. Wye-delta H- and A-extension minors of G We are ready to define wye-delta 3-sum. Definition. Suppose Gj and G 2 are disjoint and 3-connected with wye-delta subgraphs D\ and D 2 , respectively. Let Hj = Dj, with W{Hi,Di) = {uj, Cj}, and dj being the trivalent vertex of Dj, for i = 1,2. Let be adjacent to b\ in D\, and « 2 not be adjacent to 6 2 in ^ 2 - Also let 6 2 be adjacent to C2 in D 2 , and 6 1 not be adjacent to c\ in G\. We form the graph G from G\ and G 2 by deleting all the edges of the wye- deltas D\ and D2 along with their respective trivalent vertices di and ^ 2 , from G% and G 2 , respectively, and then identifying the pairs of vertices {a 1 , 0 2 ) 1 {^li^ 2 } and { c i, C2 }. Let the corresponding vertices of G be denoted by a, b and c. 138 respectively. Then [Hi,Ho) is a separation of G, w ith H2 ) = 3 and V{H\ n i / 2 ) = {a,b,c}. G is said to be a wye-delta 3-sum of G\ and Go (over D\ and D2 ) if there exists a proper 3-connected ffj-fixing wye-delta extension minor G'^ of G such that Gj- = Gj, for i — 1 , 2 . Conversely, a 3-connected graph G is said to be (wye-delta 3-sum) decompos able if there exist graphs G\ and G 2 with wye-deltas D\ and D 2 , respectively such that G is a wye-delta 3-sum of G\ and G 2 , over £ > 1 and £> 2 - (See Figure 7.14). Di G\ G2 G Figure 7,14. The graph G is a wye-delta 3-sum of G\ and G 2 By definition, a wye-delta 3-sum is a proper summing operation. We note that a wye-delta 3-sum of two graphs over a given pair of wye-deltas may be accomphshed in more than one way. The first result immediately follows from the definition. Proposition 7.3.1. If G is a wye-delta 3-sum of G\ and Gg then |y(G)| = |y(Gi)| + |y(G2)|-5 (7-1) |E(G)| = |E(Gi)|-b|E(G 2 ) | - 8 In order to prove that wye-delta 3-sum is closed on %, we need the following lemma. 139 Lemma 7.3.1. Suppose G is ^-connected with P{G) — 3 and let D be a wye-delta of G. Then D is localizable for G. Proof. Assume that D is not localizable for G. Let Y be the triad of D. Then, since Y spans D, it follows that Y is also not locahzable for G. Then, by (1) of Lemma 7.2.1, it follows that G has either a forbidden octacube minor or a forbidden augmented A'g 3 minor, with D fixed. If G has a forbidden octacube minor with D fixed then G has an octahedron minor, and this is a contradiction, by Proposition 6.7.5. If G has a forbidden augmented ATg g minor with D fixed then G has a minor, and this is once again a contradiction, by Proposition 6.7.5. ■ Proposition 7.3.2. The wye-delta 3-sum, is a closed summing operation on % . Proof. Using the notation of the definition of a wye-delta 3-sum, let G be a wye- delta 3-sum of G\,G 2 over D\ and Do. Also, let G%, G2 G %. We must show th a t G G 5^3 . By Lemma 7.3.1, it follows that D\ and £ > 2 are locahzable for Gi and G 2 , respectively. Let (Ti, /i) and {T1 J 2 ) be locahzations for D\ and £> 2 , respectively. Let 1 < Î < 2. Then 3e% G E{Ti), with adjacent vertices Uj and such that the set of T-leaves of the branches at Vi away from e, is labelled by D{. We form the ternary tree T by deleting the edge e, from Tj along with the branch at u, away from e,, for 1 < i < 2 , and joining the resulting divalent vertices by an edge k. (See Figure 7.15). The labelling I is defined by retaining the labels of the leaves of T that were inherited from and I2 . Then '^'{T,l) < 3. Therefore, G G %. ■ We conclude this section with some remarks about 3-summing. If G'l and G2 are disjoint and have triangles D\ and £> 2 , respectively, then a 3-sum G of G\ and G 2 is constructed by identifying the 3 vertices of D\ with the 3 vertices of £ > 2 in a one-to-one fasliion, and then deleting the edges of both triangles. Although 3-summing is somewhat simpler than wye-delta 3-summing, it is not a closed summing operation on As a counterexample, consider a 3-sum G of 2 octacubes, where the triangle in each octacube is taken to be its distinguished triangle. The octacube has branch-width 3, but G has branch-width 4 because G has the cube as a minor. (See Figure 7.16). 140 (71, fl) (71, f%) F ig u re 7.15. Obtaining {TJ) from {T iJJ and {T0 J 2 ) Figure 7.16. A 3-sum of 2 octacubes 7.4. The wye-delta 3-sum generators for % We begin by proving sufficient conditions for G to be a wye-delta 3-sum gen erator. Proposition 7.4.1. Suppose G is Z-connected. Then ( 1 ) If |y(G)| < 4 or \E{G)\ < 9 then G is a wye-delta 3-sum generator; (2) If |F (G )| < 8 , or G is a generator and G has no wye-delta then G is an inactive generator. Proof. (1) We prove by contradiction. Assume G is not a generator, and let G be a wye-delta 3-sum of Gi and Gg. Then, by Proposition 7.3.1, it follows that M Gi)|-b|y(G 2)|-5 > |y(G,.)| and |F(G])|-HE(G 2)|-8 > |F(G,)|, fori = 1,2. Therefore, |y(G)| > 5 and |F(G)| > 10. (2) First, suppose |jF(G)| < 8 . Then, by (1), G is a generator. Assume G is an active generator. Then there exist graphs H and K such that K is a wye-delta 141 3-sum of G and H. Then |E(G)| + \E{H)\- 8 > \E{H)\. Therefore, \E{G)\ > 8 , which is a contradiction. That every active wye-delta 3-sum generator must possess a wye-delta follows from the definition of a wye-delta 3-sum. H We define big 3-separations. Definition. Suppose {H,K) is a separation of G, of order 3. Then {H,K) is said to be a big 3-separation if \E{H)\ > 5 < \E{K)\. (See Figure 7.17). Figure 7.17. A big 3-separation {H,K) of the graph G We note that if G is planar then G has a big 3-separation if and only if G* has a big 3-separation. The following necessary and sufficient condition is useful for proving decomposabihty. Proposition 7.4.2. Suppose G is 3-connected. Then G is wye-delta 3-sum de composable if and only if G has a big 3-scparation (H,K) and G has proper 3-connected wye-delta H- and K-extension minors. Proof. This follows from the definition of wye-delta 3-sum decomposabihty. B In fight of the above results, we observe that K 4 , IF 4 , the 3-prism, the dual of the 3-prism and must all be generators for %, with K 4 , W4 and ATg 3 being inactive generators. We also see that the octacube and augmented ATg 3 graph must be generators for % because neither of these two graphs has a big 3-separation. On the other hand, although the graph W 5 has a big 3-separation, we are able to prove that IT 5 is not decomposable. 142 Proposition 7.4.3. The graph is a wye-delta 3-sum generator. Proof. Assume that is wye-delta 3-sum decomposable. Figure 7.18(a) illus trates IF5 . Hi (a) G raph W 5 (b) A big 3-separation {Hi, Hf) Figure 7.18. The graph W 5 and a big 3-separation {H\, H2 ) Suppose {H\,H 2 ) is a big 3-separation. Then \E{Hi)\ = 5 = \E{H2 )\- Also, H\ and H2 are expressible as given by Figure 7.18(b), with the vertices common to H\ and H2 being a, b and c. GU Gi (a) Figure 7.19. Two pairs (G^, Gg) of non-summands of TF 5 It suffices to show that IF 5 has no proper 3-connected wye-delta Hi extension minor G(-, for i = 1,2. If G\ and Gg are to be 3-connected then they should be as given in Figure 7.19(a), but then neither of these is a minor of IF 5 . On the other hand, if Gj and G'2 are to be proper minors of IF 5 then they should be as given in Figure 7.19(b), but then neither of these is 3-connected. B We next define the concept of a non-generator. 143 Definition. Suppose G is 3-connected and G E If G is not isomorphic to any of the eight 3-connected generators, then G is called a non-generator (for % with respect to the wye-delta 3-sum). In Section 7.3 we estabhshed that the wye-delta 3-sum is a proper closed summing operation on ^ 3 . To complete the proof of the claim illustrated by Figure 7.1, regarding the internal structure of %, we must show that every non generator is decomposable. We prove decomposabihty by using Proposition 7.4.2. Then, by induction, it follows that every 3-connected graph G of % is either a generator or is finitely generated from the five 3 -connected active generators of % via wye-delta 3-summing. For 10-edge non-generators, we give a direct proof of decomposabihty. There after, we show that every non-generator G that possesses at least 1 1 edges has a big 3-separation. The cases where G has 11 or 12 edges are tackled in Propo sitions 7.4.5 and 7.4.6, respectively. The general case, where G has at least 13 edges, is dealt with in Proposition 7.4.7. Proposition 7.4.4. Suppose G consists of 10 edges and is 3-connected. Then (1) iyG% ATs f/ien/3(G) = 3; (2 ) If G is a non-generator then G is unique and wye-delta 3-sum decompos able. Proof. ( 1 ) Since |E(G)| = 10 it foUows that G has at most 6 vertices. If G has fewer than 6 vertices then either G = or G is not simple. Therefore, G has exactly 6 vertices. Therefore, the vertex sequence of G is either (5,3, 3,3,3,3) or (4,4, 3,3, 3,3). If the vertex sequence of G is the former then G is isomorphic to IF3 , which is a generator. Therefore, the vertex sequence of G is (4,4, 3,3,3, 3). Let u and v be the two vertices of valency 4. Then u is adjacent to v, because otherwise G would have a 2-separation. (See Figure 7.20). G Figure 7.20. A 2-separation of G 144 Let g' be the graph that is obtained from G by deleting the edge that joins u and V. T hen G' has 6 vertices and 9 edges, and is regular of valency 3. Also, G' is at least 2 -connected. If G' has no triangles then G' = ATg g, and G is isomorphic to the augmented fig g graph, which is a generator. So, suppose G' contains a triangle. Then it follows that G' must have two vertex-disjoint triangles, and from this it follows that G' is the 3-prism. Therefore, G is obtained by joining two non-adjacent vertices of a 3-prism. Since G has a wye-delta, from ( 1 ) and (2) of Proposition 6.7.4 it follows that /3{G) = 3. (2) Suppose G is a non-generator. We have just proved the uniqueness of G. (See Figure 7.21). It can now be verified that G is a wye-delta 3-sum of the 3-prism and its dual. B Figure 7.21. The unique 10-edge non-generator Proposition 7.4.5. Suppose G consists of 1 1 edges. Then /3'{G) < 3 if and only if G does not have as a minor. Also, if G is 3-connected and P'{G) = 3 then G has a big 3-separation. Proof. If 0{G) < 3 then G does not have JF 5 as a minor because = 4. Conversely, suppose G does not have as a minor. If G is not 3-connected then G is a 0-, 1- or 2-sum of two of its proper minors. Therefore, by Proposition 7.4.4, it follows that /3'{G) < 3. Next, suppose G is 3-connected. We show that there are 6 possibilities for G. The result then follows because each of these admissible graphs is of branch-width 3 and has a big 3-separation. We use vertex sequences. The graph G has at least 6 vertices and no more than 7. If G has a vertex of valency 5 then the vertex sequence of G is either (5,5,3, 3,3,3) or (5,4,4,3,3,3). If G has no vertex of valency 5 then G must have a vertex of valency 4. In this instance the vertex sequence for G is (4,4,4,4,3, 3) or (4,3,3,3,3,3,3). We show that the vertex sequence of G cannot be (5, 5,3,3,3,3). If so, each vertex of valency 5 is adjacent to all the other vertices, and each trivalent vertex 145 is adjacent to exactly one other vertex of like valency. Then G has a 2 -separation, and hence is not admissible. (See Figure 7.22). Figure 7.22. The graph with vertex sequence (5,5,3,3, 3,3) If the vertex sequence of G is (5,4,4, 3,3,3) then there are two possibihties. The vertex of valency 5 is adjacent to all the other vertices, but the two vertices of valency 4, which we denote by u and n, may or may not be adjacent to each other. Let us denote the set of trivalent vertices that are adjacent to u and v by Z\a) and •Z(n), respectively. First, suppose u and v are adjacent to each other. Then |Z(n)| = \Z{v)\ — 2 and \Z{u) fl Z{v)\ = 1. Therefore, two of the trivalent vertices are adjacent to each other, but the other trivalent vertex is adjacent only to the vertices of higher valency. (See Figure 7.23(a)). Next, suppose u and v not adjacent to each other. Then \Z{u)\ = |Z(n)| = 3. This forces the 3 trivalent vertices to form an independent set. (See Figure 7.23(b)). Both these graphs are admissible. u (b) Figure 7.23. Graphs with vertex sequence (5,4,4,3,3,3) If the vertex sequence of G is (4,4,4,4,3,3) then there are two possibilities to start with, depending on whether the two trivalent vertices are adjacent to each other or not. Let us denote the two trivalent vertices by u and v. First, suppose that u and v are not adjacent to each other. Then |N(n)| — |iV(i;)| = 3 and \N{u) D N{v)\ = 2. Also, the 2 vertices of N{u) D N{v) are not adjacent to each other, but every other pair of vertices of valency 4 is an adjacent pair. (See Figure 7.24(a)). This graph is admissible. Next, suppose that u and 146 V are adjacent to each other. Let N'{u) — N{u)\v and N'{v) = N{v)\u. Then \N'(u)\ = |7V'(t;)| = 2 and N'{u) fl N'{v) — 0. The four vertices of valency 4 also have to be the vertices of a 7 ^ 4 subgraph of G. This graph has as a minor, and hence is not admissible. (See Figure 7.24(b)). u (b) Figure 7.24. Graphs with vertex sequence (4,4,4,4,3,3) Finally, suppose the vertex sequence of G is (4,3,3, 3, 3, 3, 3). We show th at there are 3 possibihties for G. Let the two trivalent vertices to which the vertex of valency 4 is not adjacent be denoted by u and v. If u is adjacent to v, let N'(u) — N(xi)\v and N'{v) — N{v)\u. Then |fV'(u)| = \N'{v)\ = 2 and 0 < |lV'(w)nlV'(u)| < 1. If |N'(u)nlV'(u)| = 1 then the vertices of G that are adjacent to at most one of u and v are the vertices of a 4-cycle. The graph G is as illustrated in Figure 7.25(a), and is isomorphic to the dual of the planar graph that we obtained when the vertex sequence is (5,4,4,3,3,3). If N'{u) n N'{v) = 0 then in order for G to be 3-connected, each trivalent vertex that is adjacent to u must be adjacent to a trivalent vertex that is adjacent to v. The graph G is as illustrated in Figure 7.25(b) and is isomorphic to the dual of the planar graph that we obtained when the vertex sequence is (4,4,4,4,3,3). If u is not adjacent to v then |lV(u)| = |iV(u)| = 3 and \N{u)r\N{v)\ = 2. The graph G is non-planar because it contains TCg g as a minor. (See Figure 7.25(c)). Each of these 3 graphs is admissible. B UV u u V (a) (b) (c) Figure 7.25. Graphs with vertex sequence (4,3,3,3, 3,3,3) 147 Proposition 7.4.6. Suppose G is a non-generator that has 12 edges. Then G has a big 3-separation. Proof. Assume that the result is false. Then, from the cential-edge-lemma it foUows th at G is the union of 3 edge-disjoint wye-deltas Zi, Z2 and Z 3 , say. We obtain a contradiction by proving that G must be isomorphic to the octacube, which is a generator. For 1 < i < 3, let V{Zi) = {a,, 6 %, c,, d,} with hi adjacent to Cj in Z,, and di trivalent in G. T hen W(G, Zj) = {uf, 6j-, c;}. Also, Of is not adjacent to either 6 ; or cf for, otherwise, G has a big 3-separation. Therefore, if z j then 1 < \V{Zi f] Zj)\ < 2. As |iy(G, Zj)| = 3, without loss of generality, suppose |F (Z i n Z ‘2 )| = 2, «1 = 02 and C] = C2- Now, {03,63,03} Ç {01,61,01,62}- Also, out of the six pairs of distinct vertices obtained from { 0 1 , 6 1 , 0 1 , 6 2 }, only 6 1 may be adjacent to 62 in Z3. T hen c?3 must be adjacent to oi, 61 and 62. Therefore 03 — 01, 63 = 61 and 0 3 — 62. Thus G is isomorphic to the octacube. (See Figure 7.26). B Cl = C2 61 = 3 Figure 7.26. The octacube Proposition 7.4.7. If G is a non-generator with at least 11 edges then G has a big 3-separation. Proof. If 11 < \E{G) I < 12 then G has a big 3-separation from Propositions 7.4.5 148 and 7.4.6, and if |£7(G)| > 13 then G has a big 3-separation by the central-edge- lem m a. B Due to Proposition 7.4.4, it suffices to consider only those non-generators that have at least 11 edges. Using the knowledge that every such non-generator G has a big 3-separation {H,K), we wish to show that G has proper 3-connected w ye-delta H and A"-extension minors. The main “problem” with showing this is that these extension minors may be not 3-connected. The next section offers a solution. 7.5. Robust separations We begin with some basic definitions. Definition. Suppose G is 3-connected and {H, K) is a separation of G such th a t K) = 3. Let V[H n K) = {a, 6 , c}. The ordered pair (a, { 6 , c}) is said to be a robust selection of attachments for H if v al//(a) > 2 and b is not adjacent to c in H. In this instance H is said to have a robust selection of attachments at (a, {6 , c}). The separation (H,K) is said to be a robust separation of G if JT and K have distinct robust selections of attachments. Figure 7.27 illustrates that the graph G has {H,K) as a robust separation. In that figure, H and K have robust selections of attachments at (a, {&, c}) and (c, {a, fc}), respectively. H Figure 7.27. The graph G has a robust separation {H, K) 149 We prove via a sequence of lemmas that a wye-delta extension minor of a subgraph at a robust selection of attachments must be 3-connected. Lemma 7.5.1. Suppose G is 3-connected and [H^K) is a separation of G such that = 3. Then H is connected. Proof. Let V{H fl K) = {a, 6 , c}. Since the result is true if \E{H)\ = 2 or \V{H)\ = 3, suppose \E{H)\ > 3 < \V{H)\. Let u e V{H)\{a,b,c} and v E V{H)\u. There are 3 internally disjoint paths from v to u, in G. At m ost 2 of these paths may have edges in K. Therefore there is a path from v to u th at hes entirely in H. Hence H is connected. B Lemma 7.5.2. Suppose G is 3-connected and {H,K) is a separation of G such that = 3 and V{H E\K) = (a,i, c}. Suppose also that (a, {6 , c}) is a robust selection of attachments for H. Then any wye-delta H-extension at (a, {6 , c}) is 3-connected. Proof. Let G\ be a wye-delta .ff-extension at (a, {b,c}) and Z be the wye-delta subgraph of G\ such th a t W{G, Z) = {a,b,c}. (See Figure 7.28). Figure 7.28. A wye-delta ff-extension Gi at (a, {6 , c}) Since the result is true if \E{H)\ — 2, suppose \E{H)\ > 3. By Lem m a 7.5.1, it foUows that H is connected. Therefore, since Z is also connected and V{H) n V{Z) 0, it foUows th a t G\ is connected. To estabUsh that Gi is 2-connected, assume G\ has a cut-vertex, t say. Then since G has no cut-vertices, it foUows th at t — d. If d is a cut-vertex then so is a, and we have already discounted this. 150 Finally, to establish that G\ is 3-connected, assume G\ has a 2 -separation, (M , N) say. We consider separately the 2 cases where Z is (without loss of generahty) a subgraph of Æ, and Z is not a subgraph of either M or N. If Z is a subgraph of N then M is a subgraph of H. Since M has at least 2 edges it follows that M has at least 3 vertices of attachment in G. At least one of these vertices of attachment is distinct from a, b and c. However, that vertex of attachment must meet an edge of K. This is a contradiction. In the remaining case Z is not a subgraph of either M or N. Assume d ^ V{M n N). Then, without loss of generahty, d ^ V{M). Choose a 2-separation {M,N) such that \E{M)\ is minimal. Now, F(Z) Ç V{N). From tliis it follows th a t V{M n N) = {b,c} and that the edge joining b and c is in M. Since M has at least 2 edges M has a vertex u, distinct from b and c, such that valj^(u) — valg(u) > 3. From this it follows that M is not minimal as chosen, which is a contradiction. Thus d e V(M nN). Let v be the other vertex of M HA. Without loss of generality, let a E V{M) and 6 , c G V{N). Let M' = M ~ d. Now, since \E{M')\ > 1 and G\ is 2 -connected it follows that a ^ v. Because a has valency at least 3 in G\ it follows that \E[M')\ > 2 . Therefore, is a 2-separation of G, and this is a contradiction. (See Figure 7.29). B a + M' V M N Gi Gi Figure 7.29. The wye-delta H-extension G i 151 Proposition 7.5.1. Suppose G is 3-connected and (H,K) is a separation of G such that = 3 and V{H HK) = {a,b,c}. Suppose also that (a ,{ 6 ,c}) is a robust selection of attachments for H. Then any wye-delta H-extension minor of G at (a, {6 , c}) is 3-connected. Proof. This follows from Lemma 7.5.2, as any iï-extension minor is isomorphic to an i7-extension. ■ Using Propositions 7.4.2 and 7.5.1 we arrive at a practical way of showing that a 3-connected graph G is decomposable. First, we must show that G has a robust big 3-separation (i7, K). Let (a, {6 , c}) and ( 6 , {c, a}) be robust selections of attachments for H and Jv, respectively. Next, we must show that G has a wye-delta 77-extension minor at (a, { 6 , c}), and a wye-delta Jf-extension minor at (6 , {c, a}). These 2 steps prove that G is decomposable. 7.6. Obtaining robust separations by pushing edges In this section we discuss ways of obtaining robust separations. Definition. Suppose G is 3-connected and (77, if) is a 3-separation of G. Sup pose also that \E{H)\ > 4 and A is an edge of 77 with incident vertices x and y, and that either x,y G U(77n7i) or val/y(z) = 1. Define H\ = G ■ {E{H)\A) and K\ = G- {E{H)UA). Then the 3-separation (77%, 77%) is said to result from push ing the edge A from 77 to K, and the transformation from (77, K) to (77%, K\) is called a push or push transformation on the separations, and may be denoted by 7T. The separations (77,7f) and (77%,77%) are called the pre-push separation and the post-push separation of vr. (Figure 7.30 illustrates pusliing the edge A when both the incident vertices of A are in U (77 fl 77)). We are interested in two kinds of pushes. These pushes are specified by their pre-push separations, and the post-push separations that they create. Definition. Suppose G is 3-connected and (77,77) is a 3-separation of G with U (77 n 77) = {a, 6 , c}. Then 77 is unsuitable for a robust selection of attachments if either a, b and c are all pendant vertices of 77, or if a, b and c are the 3 vertices 152 A / / \ \ / ( / / A -> 1 Figure 7.30. An example of a push of a triangle of H. We denote any push from {H,K) to under these hypothesis by ttj. N otation. Consider a transformation ttj from {H,K) to {H\,K\). First, sup pose a, h and c are all pendant vertices of ff, with incident edges A, B and C, respectively. Let h' and c' be the non-pendant vertices of H that are adjacent with the edges A, B and C, respectively. We note that a!, b' and c' are distinct because G is 3-connected and \E{H)\ > 4. Then a, b and c are all non-pendant vertices of K. The push ttj pushes one of the edges A, B oi C from H to K. Without loss of generahty, let B be the edge that is pushed from H to K. Then V{H\ n K\) = {a, b', c}. Also, both a and c are pendant in H\ and non-pendant in A'l, while b' is non-pendant in H\ and pendant in K\. (See Figure 7.31). 4- c' c c + c' c c + H K A'l Figure 7.31. The first instance for the apphcation of tti Next, suppose a, b and c are the. 3 vertices of a triangle of H. Let the edges of H that are incident with the pairs of vertices {a, 6 }, {6 , c} and {c, a} be denoted 153 by C, A and B, respectively. Since G is 3-connected and \E(H)\ > 4 it follows th a t H contains a vertex distinct from a, b and c. The push ttj pushes one of the edges of this triangle from H to K. Without loss of generality, let C be be the edge that is pushed from H to K. Then the vertex c is non-pendant in H\, while the vertices a and b are non-pendant in Ki. (See Figure 7.32). a B c B b A c H H i A', Figure 7.32. The second instance for the apphcation of tti The second type of push is apphcable under the following situation. Definition. Suppose G is 3-connected and {H,K) is a 3-separation of G for which both H and K have robust selections of attachments. Then {H,K) is ■unswztaèle as a robust 3-separation when = {a, 6, c), where a is adjacent to b and c 'm H b u t b and c are not adjacent in H, while b and c are both pendant in K and val/{(a) > 2. We denote any push from {H,K) to under these hypothesis by 7T2. (See Figure 7.33). N otation. Consider a transformation Trg from {H,K) to In H, the vertex a is adjacent to both b and c, while the vertices b and c are not adjacent to each other. In K, the vertex a is not pendant while the vertices b and c are both pendant. Then a, b and c are all non-pendant vertices in H. Let the edges from a, to b and c (in H), be denoted by B and C, respectively. Let the pendant edges incident with b and c (in K) be denoted by P and Q, respectively. Also, let the non-pendant vertices of K that are adjacent to P and Q be denoted by b' and c% respectively. (See Figure 7.33). 154 + C b c c + c ' H K Figure 7.33. A pre-push separation for vrg The push 773 may be accomphshed essentially (without loss of generality) in one of 2 ways. One of these ways is by pushing either B 0 1 C from H to K. The other way is by pushing either P ox Q from K to H. To see what happens when 7T2 pushes an edge from H to K, let us, without loss of generality, suppose that B is pushed by Trg. Then V{Hi r\K\) — V{H r\ I-'). The vertices c and a are non-pendant in H\ and K\, respectively. Also, c is a pendant vertex and 6 is a non-pendant vertex of K\. (See Figure 7.34(a)). To see what happens when tt2 pushes an edge from K to H, let us, without loss of generality, suppose that P is pushed by tv2 - T hen V{H\ D l i i ) = {a, b' ,c}. The vertices b' and c are pendant in Hi and /vi, respectively. The vertex c is non-pendant in and the vertices a and b' are non-pendant in K\. (See Figure 7.34(b)). + c b ^b' c c c.1 ‘ + c ' (a) (b) lU Hi /Cl Figure 7.34. Two post-push separations for 7T2 155 In the case of 7T2 an edge may be pushed from H to K or from K to H. For 7T1 however, an edge may pushed from H only if H is unsuitable with respect to 7T1 . The first result to follow characterizes non-robust separations in terms of pre-push separations. Proposition 7.6.1. Suppose G is 3 -connected and (H,K) is a 3-separation of G. Then (1) The subgraph H does not have a robust selection of attachments if and only if {H,K) is a pre-push separation for n\, with H being unsuitable with respect to vrp (2) Suppose H and K each have a robust selection of attachments. Then {H, K) is not a robust separation if and only if {H, K) or (K, H) is a pre-push separation for uo- Proof. Let V{H fl Jv ) = {a, 6 , c}. (1) First, suppose {H, K) is a pre-push separation for ttj, with H being unsuitable with respect to tti. Then either a, b and c are all pendant vertices of H, or a, b and c are the 3 vertices of a triangle of H. Therefore, H does not have a robust selection of attachments. Next, suppose H does not have a robust selection of attachments. Assume (/f, K) is not a pre-push separation for ttj, with H being unsuitable with respect to 7T]. Then, without loss of generality, the vertices b and c are not adjacent to each other in H. Therefore, the vertex a is pendant in H. Without loss of generality, by our assumption, c is not pendant in H. Therefore, a m ust be adjacent to b in H. This gives a 2 -separation of G, where the two edge-induced subgraphs of the separation intersect in the set { 6 , c} of 2 vertices. This is a contradiction. (See Figure 7.35). (2 ) We use the notation of the definition of 7T2 and Figure 7.33. First, suppose {H,K) is a pre-push separation for 7T2 . Then, from Figure 7.33 it follows that (a, { 6 , c}) is the only robust selection of attachments for H as weU as K. Hence (H,K) is not a robust separation for G. Next, suppose {H,K) is not a robust separation for G. By the hypothesis, H and K each have a robust selection of attachments. Hence it follows that they 156 H K + c Figure 7.35. A 2-separation of G must have exactly one robust selection of attachments each, and these robust se lection of attachments must be identical. Without loss of generality, let (a, { 6 , c}) be the unique common robust selection of attachments of H and K. Then a is a non-pendant vertex of both H and K, and the vertices b and c are not adjacent to each other in both H and K. Without loss of generality, suppose a is not adjacent to b in K. Then c is pendant in K. Also, c cannot be adjacent to a because otherwise G will have a 2-separation. Therefore, b must be a pendant vertex of K. Since b and c are pendant vertices of J\, it follows that these two vertices are non-pendant vertices of H. Therefore, a is adjacent to both c and b in H. Therefore, {H,K) is a pre-push separation for 7T2 - (Observe how H and K agree with Figure 7.33). Bl The second result proves the efficacy of the push operations in obtaining robust separations. Proposition 7.6.2. Suppose G is 3-connected and {H,K) is a non-robust 3- separation of G. Then a robust separation of order 3 may be obtained by performing exactly one push operation on {H,K). Proof. From Proposition 7.6.1 it follows that [H, K) is a pre-push separation of 7T1 or 7T2 . If (H,K) is a pre-push separation of ttj then perform the push tti, as prescribed in the definition. From Figures 7.31 and 7.32 it follows that that the post-push separation is a robust separation. Similarly, If [H,K) is a pre push separation of 7T2 then perform the push 7T2 , as prescribed in the definition. 157 From Figure 7.34 it follows that that this post-push separation is also a robust separation. B The third result is a useful refinement of the second. Proposition 7.6.3. Suppose G is 3-connected and has at least 11 edges. Suppose also that (H,K) is a non-robust big 3-separation of G. Then a robust big 3- separation may be obtained by performing exactly one push operation on Proof. If \E{H)\ > 5 < \E{K)\ then the result follows from Proposition 7.6.1. Without loss of generality, suppose |E(ff)| — 5. Then |F(K )| > 5. If {H,K) is a pre-push separation of m then we perform tt2 on {H,K) by pusliing an edge from K to H. Then |F(i7i)| > 5 < |£^(7vi)|, as required. Next, suppose {H,K) is a pre-push separation of tti, with H being unsuitable w ith respect to tt^. If H has more than 5 edges then \E{H\)\ > 5 < \E{K\)\. We show that H must have more than 5 edges. Assume H has exactly 5 edges. Let V{H n K) = { a , 6 , c}. First, suppose a, h and c are all pendant vertices of H. We use the notation of Figure 7.31. Then 2 of the distinct vertices b' and c' must be divalent, and this is a contradiction. Next, suppose a, h and c are the 3 vertices of a triangle of H. Then because any other vertex of H must have valency at least 3 in 77, it follows that H cannot have any other vertex. This forces H to have parallel edges, and this is also a contradiction. B 7.7. W ye-delta 3-sum decomposability of the wheel Wn, for ra > 6 Proposition 7.7.1. If n > 6 then Wn is wye-delta 3-sum decomposable. Proof. We begin by estabHsliing some notation. Let G = Wn-, and z and C denote the hub and the rim of G, respectively. Also, if u and v are distinct vertices of C, let the 2 internally disjoint paths from u to a in C be denoted by P\{u,v) and P2 {u,v), respectively. 158 u t z s w V H K Figure 7.36. The big 3-separation (H, K) of G We first show that G has a robust big 3-separation. Since n > 6 it follows th at 3u,v 6 V{C) such that P{{u,v) and P2 {u,v) have at least 3 edges (and 2 internal vertices) each. Let H be the subgraph of G that is induced by all the edges of Pi{u,v) and all the spokes that are incident with the vertices of V{Pi{u, v))\v. Also, let K be the subgraph of G that is induced by all the edges of P 2 {u,v) and all the spokes that are incident with the vertices of V{P\ {u, v))\u. Then {H,K) is a big 3-separation of G. Also, since (u, {z,v}) and {v, {z,u}) are distinct robust selections of attachments for H and K, respectively, it follows th at (H,K) is a robust separation of G. To complete the proof, it suffices to show that G has proper wye-delta. H- and /\-extension minors at {u,{z,v}) and (u, {z, «}), respectively. Since {H,K) is a big 3-separation, it follows that P\{u,v) and P2 {u,v) have at least 2 internal vertices each. Let t and s be the first 2 internal vertices of Pi(u,u), respectively, and w and x be the last 2 internal vertices of P2 {u, v), respectively. We construct a proper wye-delta if-extension minor G\ at (u, {z, u}) by deleting all the spokes that are adjacent with the internal vertices of Fg (u, u) except for the spoke that is adjacent w ith w, and contracting all the edges of Fg(?r, u) except the 2 edges from X and v to w. A a proper wye-delta A-extension minor Gg at (u, {z,w}) is constructed in a hke manner. (See Figure 7.36). B 7.8. W ye-delta 3-sum decomposability of a general non-generator We are able to prove the culminating result after 3 lemmas. We need the following definition. 159 Definition. Suppose G is 3-connected and (ff, K) is a 3-separation of G such th at V[H K) = {a, 6 , c}. Suppose also that K contains every edge of a path P from a to 6 , and all the edges that join the internal vertices of P to the vertex c. The only other edges that K may contain are the edges that join c to a and h. Then G is said to be a part-wheel H-extension and c is said to he the center of K. (See Figure 7.37). b KH Figure 7.37. The graph G as a part-wheel Ff-extension Lemma 7.8.1. Suppose G is 3-connected and {H, K) is a 3 -separation of G such that V[H n K) = {a, 6 , c}. Then exactly one of the following must occur. (i) The graph G has a wye-delta H-extension minor at (c, {a,/;}), or (ii) The graph G is a part-wheel H-extension with c being the center of K. Proof. We begin by showing the exclusivity of (i) and (ii). If (i) is true then |F(7'L')| > 4. If (ii) is tru e then there is a unique i^ath from a to 6 in K that does not have c as an internal vertex. Hence (i) cannot happen. If \E[K)\ — 3 then K is either a triad or a triangle. Therefore, (ii) holds. So, suppose \E{K)\ > 4. We focus our attention on G — c, and consider separately the cases where K — c is and is not a forest. Letting G' = G — c, H' = H — c and K' = — c, we observe that G' has [H',K') as a separation of order 2, with V{H',K') = { a ,6}. (See Figure 7.38). 160 G' a H' K' Figure 7.38. The graph G' and its separation {H',K') First, suppose K' is a forest. We prove that (ii) holds by showing that K' is an arc with ends a and b. Since \E{K)\ > 3 it follows that E{K') ^ 0. Also, since G has no pendant vertices, K' has no isolated vertices. Finally, to show that K' has no pendant vertices other than a or 6 , assume such a vertex p exists. Then, since valj^-(p) = valg'(p) > 3, it follows th at p must be joined to c by at least 2 edges. This is a contradiction. Next, suppose that K' has a cycle C, say. Then C is 3-joined to the vertices a, b and c, in K. Therefore, (ii) holds. (See Figure 7.39). ■ G raph G Minor of G Figure 7.39. The graph G and its wye-delta iî-extension minor Lemma 7.8.2. Suppose G is 3-comiected and {H, K) is a robust big 3-separation with V{H r\ K) = {w, z, 1/}. Suppose also that {w, {z,y}) is a robust selection of attachments for H . Then one of the following must hold. (i) The graph G has a proper 3-connected wye-delta H-extension minor G\ at (w, {z,y}), or (ii) The graph G is a wheel, or (in) The graph G is wye-delta 3-sum decomposable. 161 Proof. Figure 7.40 illustrates the hypothesis. The 3-connectivity of G\ in (i) follows from Proposition 7.5.1. w G H Figure 7.40. The graph G and its subgraph H Suppose (i) fails. Then, by Lemma 7.8.1, it follows that G is a part-wheel i7-extension with w being the center of K. Hence K consists of a path P from X to y w ith w being joined to all the internal vertices of P (and possibly to x a n d /o r y). Since |£'(J\)| > 5, it follows that P has at least one internal vertex, val/^(u;) > 2, and x is not adjacent to y in K. We prove that either w is not adjacent to x or m is not adjacent to y in H. Assume otherwise. Then, since w is adjacent to both x and y in H, it follows th at (w, (x, y}) is the only robust selection of attachments for H. Also, w is not adjacent to either x or y, in K. Therefore, x and y are both pendant vertices of K. Therefore, (m , {x , t/}) is the only robust selection of attachments for both H and K. This is a contradiction because {H, K) is a robust separation of G. (See Figure 7.41). X X o - w w < V y » H K Figure 7.41. A separation {H,K) of G 162 Without loss of generality, suppose w is not adjacent to x in H. If y is not a pendant vertex of H then (y, {w, z}) is a robust selection of attachments for H, and G has a proper 3-connected wye-delta -extension minor at (y, {w, %}). Also, by Lemma 7.8.1, either G has a proper 3-connected wye-delta A'-extension minor at (uj, {z,y}), or G is a part-wheel AT-extension with w being the center of H. If the former apphes then (iii) is true, and if the latter does then (ii) follows. Finally, suppose y is a pendant vertex of H. T hen w is not adjacent to y in A, for otherwise the subgraph H — y has at least 4 edges but only 2 vertices of attachment. Therefore, by the previous argument, the result is true if x is not a pendant vertex of H. So, suppose x is also a pendant vertex of H. Then both X and y are non-pendant vertices of K. Therefore, w is adjacent to both x and y in A . Hence (H,K) is a pre-push separation of ixo- This contradicts the fact th a t {H,K) is a robust separation of G. ■ Lemma 7.8.3. Suppose G is 3-connected and {H, K) is a robust big 3-separation with V{H n K) = (w, z, y}. Suppose also that w is a pendant vertex of H, and X is not adjacent to y in H. Then either G is decomposable or G is a wheel. Proof. Figure 7.42 illustrates the hypothesis. w W H Figure 7.42 The graph G and its subgraph H Suppose w' is the unique vertex of H that is adjacent to w. Then w' is distinct from both x and y. Therefore, {w, z, y} is a set of independent vertices of H. Without loss of generality, suppose x is not a pendant vertex of H. Then (z, {w, y}) is a robust selection of attachments for H. Then, by Lemma 7.8.2, 163 we may, without loss of generality, assume that G has a proper 3-connected wye-delta ^-extension at (z, {?c,?/}). Because (H,K) is a robust separation of G it follows that either (w, {x, j/}) or {jj,{w,x}) must be a robust selection of attachments for K. Then, by Lemma 7.8.2, we may, without loss of generality, assume that G has a proper 3-connected wye-delta Lf-extension at {w, {z,y}) or (y, {ui, z}), respectively. In either case G is decomposable. BB Proposition 7.8.1. Suppose G is 3-connected, has at least 11 edges, and also has a big 3-separation. Then G is wye-delta 3-sum decomposable. Proof. Suppose {H,K) is a big 3-separation of G. Without loss of generality, suppose {H,K) is not a robust separation. Then, by Proposition 7.6.3, it follows that a robust big 3 -separation may be obtained by performing one push operation on {H,K). Let V{H\ fl Lfi) = {a, 6 , c}, with (a, ( 6 , c}) and ( 6 , {c, a}) being robust selections of attachments for H\ and K\, respectively. Then, by Lemma 7.8.2 and Proposition 7.7.1, it follows that either G is decomposable, or G has proper 3-connected wye-delta H\- and Jfi-extension minors at (a, {b,c}) and ( 6 , {c, a}), respectively. In the latter case also G is decomposable. B We note that the above result is true irrespective of the branch-width of G. 7.9. Obstacles to % In this section we prove that the cube, the octahedron, A ' 5 and Vg &re the only obstacles to %. We begin by establishing that these graphs are indeed obstacles to % . Proposition 7.9.1. The cube, the octahedron, and Vg are all obstacles to Proof. From Propositions 6.7.5 it follows that Ag is an obstacle to 8 *3 , and that each of the other 3 graphs has graphic branch-width > 3. The result now follows from Proposition 7.4.5. H We must now prove that there are no other obstacles. Before proceeding with the formal arguments, we give a rough sketch of our proof. For our convenience, we introduce some terminology. 164 Definition. A graph that is an obstacle to SJ’3, but is not isomorphic to one of the 4 graphs given in Proposition 7.9.1 is called a new obstacle. We prove that there are no new obstacles via a sequence of results. Although some of these results may be true for obstacles in general, they are stated and proved only for new obstacles. After proving that every new obstacle is 3- connected and has no big 3-separation, we use Wagner’s Theorem (Proposition 7.9.3) to show that every new obstacle must be planar. Thereafter, we use Propo sition 1.2.1 and work only with new obstacles that have a trivalent vertex. So, suppose G is a new obstacle and 2 is a trivalent vertex of G. We show that there are two possibihties - either N{x) is an independent set of vertices of G, or exactly two vertices of N{x) are adjacent to each other. If the latter is true then we show that the graph obtained from G by deleting the edge of G[iV( 2 )] is stiU 3-connected. We let Q denote the rim of the wheel neighborhood of x and focus on the external bridges of Q, wliich are the bridges of Q that do not have 2 as a vertex, and do not consist of an edge joining a pair of vertices of N{x). We find 2 graphs G\ and G 2 such that G has to be a minor of one of them. The graph G\ is ruled out because/3{Gi) — 3. The graph G 2 is ruled out because although /1 (G 2 ) > 3 it contains a known obstacle as a minor, and every minor of G2 th at does not contain such a minor has branch-width 3. We are ready to begin our proof. Proposition 7.9.2. If G is a new obstacle then G has at least 12 edges, is 2 -connected and has no big 2 -separation. Proof. That G has at least 12 edges follows from Propositions 7.4.4 and 7.4.5. That G is 3-connected follows from Proposition 5.4.4, and the nonexistence of a big 3-separation follows from Proposition 7.8.1 B We state Wagner’s Theorem (without proof). Proposition 7.9.3 (W agner’s Theorem). Suppose G is a 2-connected graph that does not have as a minor. Then (i) The graph G is planar, or (ii) The graph G is isomorphic to Vg, or 165 (iii) 3U Ç V{G) such that \U\ = 3 and w(G — U) > 3. ■ Figure 7.43 illustrates possibility (iii) for G as given by Proposition 7.9.3. G Figure 7.43. The 3-vertex cut-set U of G such that ca(G — U) >3 Proposition 7.9.4. If G is a new obstacle then G is planar. Proof. From Proposition 7.9.2 it follows that |£'(G)| > 12. Also, as Vg is not a new obstacle, G % hg. If G is not planar then from Proposition 7.9.3 it follows th at 3U Ç V[G) such that \U\ = 3 and w(G — U) >3. Then, from Proposition 7.9.2 it follows that w(G — U) = 3 and each component of G — U consists of a single vertex. Also, from Proposition 7.9.2 it follows that the subgraph G[U] of G may have at most one edge. Hence G is isomorphic to lig 3 or the augmented 7 ^3 , 3 graph, both of which are generators. This is a contradiction. ■ By Proposition 7.9.4, every new obstacle is planar. Therefore, from Proposition 1.2.1, it follows that if G is a new obstacle then either G or G* must have a trivalent vertex. Hence, by passing to the dual if necessary, it is sufficient to consider new obstacles that have such trivalent vertices. Lemma 7.9.1. Suppose G is a new obstacle. Then (1) If X is a trivalent vertex of G then at most one pair of vertices of N[x) are adjacent to each other in G. (2) No two trivalent vertices of G that are adjacent to a common vertex, are adjacent to each other. 1 6 6 Proof. Follows from Proposition 7.9.2. B Figure 7.44 illustrates the excluded subgraphs for G. C G Figure 7.44. The excluded subgraphs H for a new obstacle G Lemma 7.9.2. Suppose G is a new obstacle, and x is a trivalent vertex of G. Then (1) The subgraph G[7V(z)] has at most one edge. (2) Suppose G[A^(x)] has an edge A. Then G — A is ^-connected. Proof. (1) Follows from Lemma 7.9.1. (2) Let N{x) = {a, 6 , c}, and, without loss of generahty, let the edge A join the vertices b and c of N{x). From Proposition 7.9.2 and (3) of Lemma 1.2.2 it follows that X has a wheel neighborhood. Figure 7.45 illustrates the two possible types of wheel neighborhoods of x. Figure 7.45. The possible wheel neighborhoods for x Assume G — A is not 3-connected. Then G — A is non-separable and has a 2-separation (H,K), say. Now, without loss of generality, h E V{H)\V{K) and c G V{K)\V{H), for otherwise G itself would have a 2 -separation. Therefore, X G V{H n K). Let v be the other vertex of V{H fl K) and, without loss of generality, let a G V[K). Also, let H' be the subgraph of G obtained from H by deleting the edge joining x and h. (See Figure 7.46). 167 H' V Figure 7.46. The new obstacle G and its subgraph H' Then H' must consist of a single edge, that edge joining b and v. Therefore, the trivalent vertices x and h are adjacent to each other and also to the vertex c. This contradicts Lemma 7.9.1. Therefore, G — A is 3-connected. (See Figure 7.47). ■ C G Figure 7.47. The new obstacle G with contradictory forbidden subgraph N otation. Suppose G is a new obstacle and a: is a trivalent vertex of G. Let N{x) = {a, 6 , c}. From (3) of Lemma 1.2.2 it follows that x has a wheel neigh borhood. Let Q denote the rim of this wheel neighborhood and Lj, Lg and Lg denote the paths in Q, from b to c, from c to a and from a to b, respectively. Also, let Vi — V{Li), and let V- be the set of internal vertices of Li, for 1 < i < 3. From Lem m a 7.9.2 it follows th a t G[iV(a;)] has at m ost one edge. If G[A'(x)] has an edge then let that edge be denoted by A, and let it join the vertices b and c. (See Figure 7.48). We prove a result about the wheel neighborhood of a trivalent vertex of an obstacle. 168 Figure 7.48. The paths L\, and L 3 in Q Lemma 7.9.3. Suppose G is a new obstacle and x is a trivalent vertex of G. Then (1) If G[jV(z)] is edgeless then each of the paths L\, L 2 and L 3 has at least one internal vertex. (2) If G[7V(z)] has an edge A then each of the paths L\, L 2 and L 3 in G — A has at least one internal vertex. Proof. ( 1 ) This follows because G[A^(x)] is edgeless. (2 ) From Lemma 7.9.2 it foUows that G — A is 3-connected. Therefore, from (3) of Lemma 1.2.2 it follows that x has a wheel neighborhood in G — A. The result now foUows because G[Nq_j^{x )] is edgeless. ■ Definition. Suppose G is a new obstacle and r is a trivalent vertex of G. An external bridge of Q is a bridge B ol Q such that x ^ V{B) and B does not consist of an edge joining two of the vertices of N{x). Lemma 7.9.4. Suppose G is a new obstacle and x is a trivalent vertex of G. Suppose also that B is a external bridge ofQ. Then W{G.,B) is not contained in any one of the vertex-sets Vi, V2 or V3, but it is contained in the union of some 2 of these vertex-sets. Proof. L et W — W{G,B). Then W Ç V\ U V2 li V^. We see that W is not contained in any one of V\, V2 or V3 because if so then G would have a 2- separation contradicting Proposition 7.9.2. Also, if IF fl 0, for 1 < * < 3 then W contains a subgraph that is isomorphic to a subdivided cube, which is a contradiction. (See Figure 7.49). Therefore, such that VF fl Vj = 0. B 169 Figure 7.49. The new obstacle G with a contradictory subdivided cube Definition. Suppose G is a new obstacle and zisa trivalent vertex of G. Sup pose also that B is an external bridge of Q, and let W = W{G,B). T hen B is said to be an a-bridge, b-bridge or c-hridge if W Ç Vg U V 3 , IT Ç V3 U Vi or TT Ç Vi U V2 , respectively. Suppose B is an a-bridge, and b'^c' E W such that b' e V3 and c' E Vg. Then b' and c' are called the extreme feet of B if there is no vertex of attachment of B strictly between b' and 6 in Tg, and there is no vertex of attachment of B strictly between c' and c in L 2 . The extreme feet of b- and c-bridges are defined analogously. From Lemma 7.9.4 it follows that every external bridge of Q is an a-bridge, 6 -bridge or c-bridge. We prove a lemma about the external bridges of Q. Lemma 7.9.5. Suppose G is a new obstacle and x is a trivalent vertex of G. Also, let r E {a, 6 ,c}. Then (1) The circuit Q has at most one r-bridge. Suppose the circuit Q has an r-bridge B. Then B consists of a single edge neither of whose incident vertices is adjacent to x. (2) Each of the paths L\, and L 3 has at most 2 internal vertices. Proof. F irst, suppose G[iV(a;)] is edgeless. (1) Without loss of generality, let r = a. We first show the existence of an a-bridge of Q that consists of a single edge, wliich is adjacent to neither b or c. Among the a-bridges of Q, choose a bridge B such that b' is closest to b in I/ 3 , and subject to this condition c’ is closest to c in L 2 , where b' and c' are the extreme feet of B, as defined earfier. 170 Let the path in Q from h' to d that contains a as an internal vertex be denoted by Q [ h ' Also, let H be the union of Q[h\a^c'] and all the external bridges of Q whose vertices of attachment are contained in V{Q[b',a^c']). (See Figure 7.50). H Figure 7.50. The wheel neighborhood of x and the subgraph H Then L7) = {b',a,c'}. Therefore, \E{H)r\E{Q)\ > 2. Also, from Lem m a 7.9.3 it follows that |(£^(iï))^| > 5. Therefore, from Proposition 7.9.2 it follows th at \E{H)\ < 4. Therefore, V{H) Ç V{Q), and from this it follows that V{G) = X U V{Q). Hence, B consists of a single edge with incident vertices b' and c'. In order to prove that b' ^ b and d ^ c, without loss of generality, assume b' — b. Then from Lemma 7.9.3 it follows that L 3 has an internal vertex bo, say. By the planarity of G, the external bridge of Q whose edge is incident with 6 g, must be an a-bridge. Therefore, \E{H)\ > 5 which is a contradiction. A similar argument shows that d ^ c. In order to prove the uniqueness of B as an a-bridge of Q, we assume there is another a-bridge, say B\^ with extreme feet 5] and c\, where Iq G P 3 and c\ G Vo. By choice of B, either b\ is an internal vertex of the path in Q from b' to a, or c\ is an internal vertex of the path in Q from d to a. Therefore, \E{H)\ > 5, which is a contradiction. (2) Each internal vertex of L{ is incident with the single edge of an external bridge of Q, for 1 < i < 3. From Lemma 7.9.4 and part (1) of this lemma it foUows th at Q may have as external bridges at most one a-bridge, one 5-bridge and one c-bridge. The resrdt now foUows from this. 171 Next suppose G[iV(x)] has an edge. Then the vertices b and c of N{x) are joined by an edge A. From Lemma 7.9.2. it follows that the graph G' = G — A is 3-connected. Since ( 2 ) follows from ( 1 ) it suffices to prove (1 ). In order to prove the assertions about a-bridges when G[N(a:)] is edgeless, we first defined H to be the union of Q{h',a,c'\ and all the external bridges of Q whose vertices of attachment are contained in V{Q[h',a^c'\). The proof then depends on the fact that \{E{H)y\ > 5. Since A is not an edge of H it follows th at \{E{H)Y\ > 5, if G[N(x)] has an edge. Therefore, the assertions about a-bridges are true. In order to prove them for 6 - and c-bridges, we once again observe that the edge A is not an edge of H, where H is defined analogously for 6 - and c-bridges, respectively. Therefore, the result follows. B Proposition 7.9.5. The set of obstacles to % consists of the cube, the octahe dron, and Vg. Proof. It suffices to prove that there is no new obstacle. From Lemma 7.9.5 it follows that Figure 7.51 gives two minor maximal graphs Gi and G 2 for a new obstacle. These graphs are minor maximal in the sense that any new obstacle G may be obtained from them by deleting the edges corresponding to external bridges and suppressing the resulting divalent vertices, or by contracting the edges between internal vertices of L\, or L 3 , or by performing both these types of operations. (a) G, Figure 7.51. Two minor maximal graphs G\ and G 2 for a new obstacle 172 We show that no minor of G\ may be a new obstacle, by proving that (3{G\) < 3. Let Ea consist of the 5 edges that are adjacent either to a, or to the neighbour of a that is a vertex of L 3 . The subsets Ei, and Ec of E{G\) are analogously defined, observing the cychc order of a, h and c. T hen {Ea, Ef,, Ec} is a partition of E{G\), and Ea, -Bfc and Ec, each consists of 5 edges. Hence by Proposition 6.3.2 the sets Ea, Ef, and Ec are all completely 3-separable. Therefore, from Proposition 2.2.4 it foUows th at P{Gi) < 3. The graph Go cannot be a new obstacle because it contains the octahedron as a proper minor. In fact, the graph obtained from G2 after contracting all the edges between internal vertices of L^, L2 and L 3 also cannot be a new obstacle because this graph also contains the octahedron as a proper minor. So the only possibihties for a new obstacle are the graphs obtained from G2 by deleting one or more of the external bridges of Q and suppressing the resulting divalent vertices. By symmetry, it suffices to consider what happens when the a-bridge and the 6 -bridge of Q are deleted. Let us denote these 2 m inors by Ggq and G2b, respectively. (See Figure 7.52). (a) G 2 a (b) G 26 Figure 7.52. The minors Gga and G 2 6 of Gg We show that no minor of Ggg may be a new obstacle by proving that f3{G2a) < 3. Let Ex be the subset of E{G2a) that consists of the 3 edges that are incident with I, and the edge A. Let Ejy be the union of the edges of E{G2a)\Ex th at are incident with 6 and the neighbours of 6 . Let Ec be the union of the edges of E{G2a)\{Ex U El,) that are incident with c and the neighbours of c. Then 173 {Ex,Ei),Ec} is a partition of E{Goa) with \Ex\ = \Ec\ = 4 and \Ei,\ = 5. Hence, by Proposition 6.4.3, the sets Ex, Ejj and Ec are all completely 3-separable. Therefore, from Proposition 2.2.4, it follows that /?((? 2 a) ^ 3. We complete our proof by showing that no minor of G2b may be a new obstacle. Let Ex be as defined in the previous paragraph. Let Eq be the union of the edges of E[G2 b)\Ex that are incident with a and the neighbours of a. Let Ec be the union of the edges of E{G2b)\{Ex U Ea) that are incident with c and the neighbours of c. T hen {Ex, Eq, Ec} is a partition of E(G2b) w ith [ÆJxl = \Ec\ = 4 and \Ea\ = 5. Hence, by Proposition 6.4.3, the sets Ex, Ea and Ec are all completely 3-separable. Hence, from Proposition 2.2.4, it follows that f^{G2b) 5 3. Therefore, no minor of may be a new obstacle. Therefore, there is no new obstacle. ■ CHAPTER VIII The Lower Ideal ^ 3 8 .1 . Introduction In this chapter we discuss the internal as well as the external structure of ^ 3 . In addition to the 1- and 2-summing which we defined in Chapter 5, we introduce 5 more summing operations for binary matroids. One of the latter, which is known as the H{)-sum, corresponds to the wye-delta 3-sum that was defined for graphs in Chapter 7. Unhke for 0 3 , we will not be able to prove that . ^ 3 is finitely generated, or that ^ 3 has a finite number of obstacles, up to isomorpliism. Results such as these, which we strongly beheve are true, wiU be stated as conjectures. The output of the computer program which is given in Appendix A shows some evidence in support of our conjectures. For reasons given at the end of Chapter 5, we shall only consider binary ma troids that are 3-connected. Also, we are only interested in binary matroids that either belong to ^ 3 , or are obstacles to this ideal. The computer program generates an isomorphic copy of every such non-regular binary matroid that has at m ost 1 2 elements. Also, the matroids generated by the program are pairwise non-isomorphic. Some output from the program is referred to, and also presented in this chapter. An explanation of the output-format is given at the beginning of Appendix A. Matroids of the form Ni where i > 0, are from Appendix A. AU fists and tables mentioned are also from Appendix A. In the next section the concepts of big 3-separation and 4-circuit/bond 3- separation are defined. Using these, we then define the 5 summing operations 174 175 mentioned above. In Section 8.3 we come across our first conjecture wliich states that every 4-structure of a matroid of ^ 3 , is locahzable. Using this conjecture we prove that all the summing operations are closed on ^ 3 . In Section 8.4 we discuss regular matroids, presenting the results of Seymour [5] and applying them to ^ 3 . We also give the computer program outputs for R\q and R i 2 - Generators for ^ 3 are discussed in Section 8.5. A conjecture for decomposabihty of matroids in ^ 3 appears here. We find sixteen 3-connected generators, prove that each is either a graph or a graft, and illustrate with figures. Obstacles to ^ 3 are discussed in Section 8 .6 . Here we conjecture that the fist of obstacles given by us, is complete (up to isomorphism). We find 10 obstacles, prove that each is either a graph or a graft, and illustrate with figures. We conclude this section by estabhshing some notation. AU matroids are binary and 3-connected. The symbol M with or without subscripts and/or su perscripts denotes a matroid. Considering that the rank function of M is denoted by p, as stated in Chapter 1, the rank functions of M{ and M[ are denoted by pi and Pj-, respectively. The foUowing notation is used for row addition in matrices. N o ta tio n . Suppose K is a matrix. We denote the row operation on If, of adding the a*'^,..., 6 ^^ rows to the row by Ra • -f iîj 4 - Rc —>• Rc- In the above description of row-addition, when deahng with a matrix of the form Rep(M), we let Ri denote the row of Rep(M). We recaU that Rep(M) is obtained from Rep(M) by deleting the 1®’’ row of the latter. 8 .2 , Five summing operations We begin by defining big 3-separations and 4-circuit/bond 3-separations. Definition. Suppose (A, B) is a separation of M such that ^(A, B) — 3. Then: (1) If |A| > 5 < |H| then (A, H) is said to be a big 3-separation of M . Abig 3-separation is denoted by ‘BTS’ in Appendix A. (2) If A is a 4-circuit/bond and \B\ > 4 then (A, R) is said to be a 4-circuit/bond 3-separation of M. A 4-circuit/bond 3-separation is de noted by ‘FCBTS’ in Appendix A. 176 Using the concept of a big 3-separation and the three 8 -element 3-connected binary matroids, M ( 1 V4 ), and A^4 , we define 3 summing operations. Definition. Let M, M; and Mr be 3-connected binary matroids such that |5(M/)| < |5(M )| > |5(Mt-)|. Also let Xg be a 3-connected binary matroid that consists of 8 elements. Then X g = M{W 4 ) or X g = Xg or Xg = X 4 . We define M to be a 3-sum of M/ and Mr of type Xg, or an Xg-sum of M/ and Mr, if the following conditions are satisfied. (1) The matroid M has a big 3-separation (A,B). (2) There exist A' ^ A and B' C B, with |A'| — 4 = |B'|, and there exist sets of minoring operations ûi and Û2 such that û"i (respectively Û2 ) con sists of |A\A'| (respectively, |5\5^1) elements, with each element of <5’] (respectively, ^ 2 ) being a minoring operation on a unique element of A\A' (respectively, B\B'). (3) If M[ , M 2 and Mg are the proper minors of M obtained by performing the operations of and û'i U ^ 2 , respectively, then Mj,M^ and Mg are all 3-connected matroids, with M \ = M;, M^ = Mr and Mg = Xg. If M is a 3-sum of type Xg (of M/ and Mr) then M is also said to be Xg- decomposable or of decom,position type Xg or simply decomposable. Also M; and Mr are called Xg-siimmands of M . In the above notation, when X g = M (W 4 ), we replace M{W 4 ) by W 4 . Figure 8.1 gives a pictorial representation of an Xg-sum. We need the following lemmas in order to estabhsh the size and rank of an Xg-sum in terms of the sizes and ranks, respectively, of its summands. Lemma 8.2.1. Suppose {A,B) is a separation of M and B' Ç B. Also, suppose M' is a minor of M such that S{M') = AiJ B' and ^(M% A, B') = ^(M , A, B). Then p{A) = p'{A). Proof. By induction it suffices to prove when = 1. Then p{B) — 1 < p'{B) < p{B) and p{M) — 1 < p'{M') < p{M), w ith p'{M') — p{M) — 1 only if /(B ) = p(B) - 1. 177 B M AB B' M' %8 M[ = Ml F ig u re 8 .1 . Pictorial representation of an Xg-sum, with relevant minors Firsc suppose p'(M ') = p{M). Then since p'(yl) = p(A)-|-p(B)—p'(B) it follows th at p’[A) = p{A). Next suppose p'{M') = p{M) — 1. Then p'{B) = p{B) — 1. Therefore, since p'{A) = p{A) -f p{B) — p'{B) + p'(M ') — p{M) it follows that p'{A) = p{A). B Proposition 8 .2 .1 . Suppose M is an Xg-sum of Mi and Mr as defined. Then |5'(M)| = |.9 (Mf)| + |g (M r)|-l p{M) = pi(Mi) 4- pr{Mr) — 4 Proof. The equation for |5(M )| foUows immediately from the definition. By Proposition 1.7.6 it follows that ^{M[,A,B') < ^{M,A,B) > ^(Mg, B). Then, since both M[ and Mg are 3-connected f,{M[,A,B') = f,{M ,A ,B ) = ^(Mg, B). Therefore from Lemma 8.2.1 it follows that p'i{A) = p{A) and p'oiB) = p{B). Also, from (b) of Proposition 6.2.3 it follows that p\[B ’) = Pg(A') = 3. Therefore, p\{M[) = p\{A) + p\{B') — 2 = p{A) -f 1. Similarly, 178 p'oiM^) = p{B) + 1. Therefore, since M/ = M\ and Mr = Mo it foUows th at Pli^l) + Pr{Mr) - 4 = p{A) + p{B) — 2 — p{M). B We note that W 4 is a weU-known graph. The matroid TVg, which is discussed further in Section 8.5, is a graft (See Figure 8.11 (a)). On the other hand, is neither a graph nor a graft. In fact A ^4 = AG{3,2), and the binary representation of A^4 is given below in the form Rep(# 4 ). / I 2 4 8 14 13 11 1 0 0 0 0 1 1 (8-2) Rep(A^ 4 ) I) 0 1 0 1 1 0 1 1 1 0 1 \ 0 0 1 1 1 0 / The matroid # 4 is strictly self dual. Figure 8.2 iUustrates 2 geometric represen tations of this matroid. 1 2 F ig u re 8 . 2 . Two geometric representations of # 4 = AG{3,2) Using the concept of a 4-circuit/bond 3-separation we define 2 more summing operations. Definition. Suppose M and Mr are 3-connected binary matroids such that \S{M)\ > \S{Mr)\ > 7. Also, suppose that M has a 4-circuit/bond 3-separation (A,B). Then: (1) We define M to be a Fano sum, or Fj-sum, of F-j and Mr if the foUowing conditions are satisfied. 179 (a) There exist A' ^ A and B' ^ B w ith |A'| — 3 = \B'\, and there exist sets of mincring operations û’i and û"2 such that û"i consists of a single minoring operation on the element of A\j4^, and ^ 2 consists of \B\B'\ minoring operations with each element of Û2 being an operation on a unique element of B\B'. (b) If M[ and M 9 are the proper minors of M obtained by per forming the operations of Û2 and respectively, then M[ and M 2 are both 3-connected. Also, M[ = Fj and M^ = Mr w ith both M[ ■ B' and • A' being 3-circuits. (2) We define M to be a Fano-dual sum, or Fj-sum, of Fj and Mr if the following conditions are satisfied. (a) This is the same as condition (a) of (1). (b) If M[ and M 9 are the proper minors of M obtained by per forming the operations of &2 and û\, respectively, then M[ and Mg are both 3-connected. Also, Mjj = Fj and Mg = Mr with both M j ■ B' and Mg • A' being 3-cocircnits. If M is a Fano sum (respectively, Fano-dual sum) ofFj (respectively, F j) and Mr then M is said to be Fano decom-posable (respectively, Fano-dual decomposable) or sim ply decomposable. Also Fj (respectively Fj) and M,. are said to be the F-j- summands (respectively, F^-sum,mands) of M. If M is either Fano or Fano-dual decomposable then M is said to be of decomposition-type F-j. The matroids Fj and Fy appear as Ni and N 2 , respectively, in Appendix A. Figure 8.3 gives a pictorial representation of a Fano-sum as well as a Fano-dual sum. We find the size and rank of a Fano sum (respectively, Fano-dual sum) M in terms of |5"(Mr)| and p{Mr), respectively. Proposition 8.2.2. Suppose M is either a Fano sum or a Fano-dual sum as defined above. Then |5(M)| = |5(Mr)| + 1. Also, (1) If M is a Fano sum then p{M) = pr{Mr) -f 1. (2) If M is a Fano-dual sum then p{M) = pr{Mr). 180 B M B B A B' B' M M [ = F-, (b) Fano-sum (c) Fano-dual-sum Figure 8.3. The matroid M, a Fano-sum and a Fano-dual-sum Proof. The equation for |5(Af)| foUows immediately from the definition. We conclude the proof for rank. From the definition, it follows that p{B) = p[M) — l. Also since .4% 19) < A, B) and is 3-connected, it follows that ^{M!),A',B) = ^{Ad,A,B). Then, from Lemma 8.2.1 it follows that p^iB) — p{B). Therefore, p(M) = p{Mr) — p'^i-^) + 3. Since p'^iA') = 2 if M is a Fano sum , and p'^i^) = 3 if M is a Fano-dual sum, the result now follows. ■ 8.3. An im portant consequence of the localizability of 4-structures We need the following lemma. Lemma 8.3.1. Suppose M is a 3-connected binary matroid and {A,B) is a separation of M such that |A| > 4 < \B\ and ^{M ^A ,B) = 3. Also, suppose a ! ^ A and B' ^ B such that \A’\ > 3 < \B'\. Let M[ and be the 3-connected minors of M such that S{M[) = Ali B' and S (M ^ — A' \J B . [See Figure 8.4). 181 Then, if /3(Mj) = 3 = and B' and A' are localizable for M[ and M^, respectively, then j3{M) = 3. B M B M[ Figure 8.4. Pictorial representations of M, Mj and M^ Proof. Let [Ti,li) and Zg) be localizations for B' (in M j) and A' (in Mj,), respectively. Let 1 < Z < 2. Then 3/j S F(T)) with adjacent vertices x{ and yj such that the set of Tj-leaves of the branches at y\ away from /i is labelled by B', and the set of To-leaves of the branches at away f^ is labelled by AL Since Mj and Mj are both 3-connected, it follows that î/>(Mj,Tj,Zj,/;) = 3. We form the ternary tree T by deleting the edge /; from Tj along with the branch at yi away from f{, for 1 < Z < 2, and joining the resulting divalent vertices by an edge k. The labeUing Z : L[T) — > S{M) is defined by retaining the labels of the leaves of T that were inherited from Zj and Z 2 . T hen ij)[M,T,l,k) = ^(M , A, B) = 3. (See Figure 8.5). 182 (T,l) (31, (i) (72,(2 Figure 8.5. Obtaining (T,/) from and (Tg, ^ 2 ) It suffices to show that if e G E {T )\k then ip{M, T, I, e) < 3. W ithout loss of generality, we may suppose that e is a non-pendant edge of the endtree of k whose T-leaves are labelled by B. We prove this as follows. The minor of M was constructed by performing a set Û of |A\A'| minoring operations - one on each element of A\A'. We re construct M '2 by starting with M, and performing one operation of <5" at a time. Also, at the same time we start with (T,l) and construct a branch decompo sition (T2 , (2 ) of M '2 as follows. Whenever we perform a minoring operation as described above, we change the branch decomposition so that we obtain a branch decomposition of the resulting minor. If the minoring operation performed was the deletion or contraction of an element x then the leaf labelled by x along with its pendant edge is deleted, and the resulting divalent vertex is suppressed. From our construction it follows that \P(%2 , ^2 ) — ^(^2,^2) — 3- Suppose Z\ and Z 2 are successive minors in our construction, with Z 2 being a m inor oi Z\. Suppose also that and ((72, m 2 ) are the branch decom positions of Z\ and Z 2 that were obtained by our construction. By induction, it suffices to prove that ^ (^ 2 ,(72,m 2 ,e) > ^(Zi,(7i,mi,e). Let x be the element of S {Z i)\S {Z2 )- By passing to the dual if necessary, we may, without loss of generahty, assume that x was deleted from Z\ in order to construct Z 2 . Suppose 1 < ( < 2. Let Qg , be the subset of S{Zi) th at labels the (7,-leaves of the end-tree of e that does not contain k as an edge, and let P^ i — S{Zi)\Q(,^i. 183 Also let Pf. j be the subset of S{Zi) that labels the {7j-leaves of the end-tree of k that does not contain e as an edge, and let Qi- i = S{Zi)\Pf. i. Then (8-3) Qe,\ = Qc,2 , Pe,2 ^ P e ,l V and Pk,2 P k,lV (See Figure 8.6). F ig u re 8 .6 . The branch decomposition ([/%,m%) Since Z-y is a restriction of we may denote the rank functions of both Z2 and Zi by the same symbol a (say). Then ^(Z,, f/j, m,, k) = a{Pf.^i) -f (r{Qk^i) ~ cr{Zj) -f- 1. Because ^(M , T, 1, A;) = 3, and k corresponds to the edge / 2 of Tg where ^(M g, Tg, ^2, ^ 2 ) = 3, it follows that r/^{Zi,Ui,mi,k) — %/)(Zg, f/g, mg, Ic). Therefore, (8-4) <^{Pk,iV) - <^{Pk,i) + 4 ^ 1 ) - <^(^2 ) = 0 Now, (8-6) î/'(Z2,l72,mg,e) - 7j^{Zi,Ui,mue) = cr{Pe,2 ) ~ o-{Pe,i) + (t{Zi) - o-(Zg) = <^{Pe,lV) - (^{Pe,l) + o-(^l) - 4 ^ 2 ) Assume ^(Zg, f/g, mg, e) < rl>{Zi,Ui,mi,e). Then, since cr(Zi) > u(Zg) > (t (Z i ) — 1 and Pk^l Ç Pg 1 , it follows that x must be a coloop of Z\ ■ Pf- i also. Therefore, a{Pk^l\x) — a(P^j) — 1. This contradicts equation 8-4. ■ 184 The following conjecture helps us to prove that the 5 summing operations are closed on ^ 3 . Conjecture 8.3.1. 7 /M G ^ 3 and F is a 4-structure in M then F is localizable for M . Consequence 8.3.1. The 5 summing operations are closed operations on ^ 3. Proof. For W 4 -, 7V3 - and 77^-sums the consequence follows from the validity Conjecture 8.3.1 and Lemma 8.3.1. Next, suppose M is a Fano sum (respectively Fano-dual sum) as defined in Section 8.2. Then B' is localizable for Fj (respectively F f). By the vahdity of Conjecture 8.3.1, it follows that A is locahzable for M. Therefore, A' is locahzable for Mr. Now the consequence follows from Lemma 8.3.1. B Conjecture 8.3.1 is analogous to Lemma 7.3.1 for graphs, which was proved us ing Lemma 7.2.1. In that lemma we proved that the octacube and the augmented 7i3,3 graph are the minimal graphs in which non-localizable 3-structures occur. Appendix A gives some information about non-locahzable 3-structures in non regular binary matroids. Table A1.4 gives the distribution of such 3-structures, while List A1.6 hsts all the matroids in which such structures occur. 8.4. Regular matroids We begin with the definition of a regular matroid. Definition. A m atroid M is said to be a regular matroid if it is representable over every field. From Tutte [6,7] we have the following very important characterization of regular matroids. Proposition 8.4.1. A binary matroid is regular if and only if it does not contain cither Fj or F f as a minor. 185 From this result it follows that if M is regular then the dual of M and every minor of M are also regular. It is also well known that all graphic and cographic matroids are regular. We prove a result which excludes 4-circuit/bonds from almost all regular ma troids. Proposition 8.4.2. Suppose M is regular, ^-connected and has at least 7 ele ments. Then M has no A-circuit/bond. Proof. Assume M has a 4-circuit/bond, A (say). Let B = S{M)\A. We prove the result separately, for |5'(M)| = 7 and \S{M)\ > 8 . First, suppose \S{M)\ = 7. Then 3 < p{M) < 4. Therefore, either M S Ty or M = F j , which contradicts Proposition 8.4.1. Next, suppose |5(M )| > 8 . We first show that B must be an independent set. Assume otherwise. Then B contains a circuit C w ith \C\ > 3. Let M' be the minor of M that is obtained by deleting all the elements of B\C and contracting all but 3 elements of C. T hen M ' = Fj, which contradicts Proposition 8.4.1. Since B is an independent set, p{B) — \B\. Therefore, from (b) of Proposition 6.2.3 it follows that p{M) = |B | + 1. Hence p*{M) = 4 + \B\ - (|H | 4 - 1) = 3. Therefore, M is not cosimple, and this leads to a contradiction. ■ For decomposable regular matroids we have the following result. Proposition 8.4.3. Suppose M is regular 3-connected and M E Suppose also that M is decomposable. Then M is W^-decomposable. * Proof. This result follows from the fact that M cannot have N 3 , N 4 , Fj or F- as a minor, because each of these 4 matroids is non-regular. SB Two regular matroids feature prominently in the decomposition theory estab- hshed by Seymour [5]. They are Fp) and F 1 2 . The matroid Fp) consists of 10 elements and is of rank 5. Treating it as a binary matroid, we may express 1 8 6 R ep(i?io) aa foUows. /I 2 4 8 16 19 25 28 14 7 \ 1 0 0 0 0 1 1 0 0 1 0 1 0 0 0 1 0 0 1 1 (8-6) Rep(i?io) = 0 0 1 0 0 0 0 1 1 1 0 0 0 1 0 0 1 1 1 0 V 0 0 0 0 1 1 1 1 0 0 / of 1 2 elem ents and is of rank 6. Treating it as a matroid, we may express Rep(i?i9) a-s follows. /I 2 4 8 16 32 7 1 1 17 34 52 56 \ 1 0 0 0 0 0 1 1 1 0 0 0 0 1 0 0 0 0 1 1 0 1 0 0 (8-7) Rep(Ri2) = 0 0 1 0 0 0 1 0 0 0 1 0 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 0 1 0 0 0 1 0 1 1 V 0 0 0 0 0 1 0 0 0 1 1 1 y th at R i 2 is a graft, as shown in Figure 8.7. Figure 8.7. The matroid R \2 as a graft Both i?io and i?,p 2 are 3-connected and self-dual, although neither of them is identically self-dual. The next result shows that Rjo plays no role in the internal structure of ^3 . Proposition 8.4.4. The regular matroid is an obstacle to 6 0 3 . Proof. The matroid 3-connected and has no 4-structure. The result now follows from Proposition 6.5.2. H 187 To further our understanding of R \2 it is necessary to transform the matrix Rep(i?,i2), as given by equation 8-7, using column permutations and elementary row operations. We begin with some column permutations. ^4 7 52 8 11 56 2 32 34 1 16 17 \ 0 1 0 0 1 0 0 0 0 1 0 1 0 1 0 0 1 0 1 0 1 0 0 0 (8-8) Rep(i7i2) = 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 1 1 V 0 0 1 0 0 1 0 1 1 0 0 0 / t each of the first 2 sets of 3 columns forms a triad, and that each of the last 2 sets of 3 columns forms a triangle. We next perform R '2 + R i R \ and Rq + R^ R^, to obtain the following. /4 6 36 8 10 40 3 48 51 1 16 17 N 0 0 0 0 0 0 1 0 1 1 0 1 0 1 0 0 1 0 1 0 1 0 0 0 (8-9) Rep(i?i 2 ) = 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 1 \ 0 0 1 0 0 1 0 1 1 0 0 0 / Fiiicdly, we permute the rows of the above matrix; letting the 4^^, 2"^, 6*'*', 1®*^ and 5*'^ rows of that matrix be the first six rows, respectively, of the new matrix shown below. (8-10) /I 5 9 2 6 10 20 40 60 16 32 48 \ 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 R ep(i?i2) - 0 1 0 0 1 0 1 0 1 0 0 0 0 0 1 0 0 1 0 1 1 0 0 0 0 0 0 0 0 0 1 0 1 1 0 1 V 0 0 0 0 0 0 0 1 1 0 1 1 J We assert the following. Proposition 8.4.5. The regular matroid R \2 consists of 2 disjoint 6 -struetures A and B such that M ■ A = M[K2a) M ■ B = M{C- 2 ) ® M{C^). Also (3{R\2) = 3. Proof. We use the matrix representation of R 1 2 , as given by equation 8-10. Let A = {1,5,9,2,6,10} and B = {20,40,60,16,32,48}. Then p(A) = p(B) = 4. 188 Therefore, {A^B) is a 3-separation of R\ 2 - Since neither A nor B contains a 4 - structure, from Proposition 6.4.2 it follows that both A and B are 6 -structures. Therefore, M ■ A ^ -^ (^ 2 ,3 ) and M - B ^ M (C g) © M{C^). From Proposition 6.4.3 it follows that (3 {R\2 ) < 3. Since R \2 is not a series- parallel graph, it follows that 0{Ri2) = 3. H Inputting Rio and R \2 to the computer program produces the following re spective outputs. Riq: (5,10) P Dual=i?io Bits': 7 14 28 25 19 /? = 4 # of Cocys: (0,15,0,15,0,0,0,1); # of Cys: (0,15,0,15,0,0,0,1); # of R-psorbs=3;^ of C-psorbs=l; # of Bases—162; R n - (6,12) P Dual=Bi2 Bits': 7 11 17 34 52 56 /? = 3 DBG # of Cocys: (2,9,12,13,18, 6 ,0,3,0,0); #of Cys: (2,9,12,13,18, 6 ,0 ,3 ,0 ,0 ); ^ of R-psorbs=10; C-psorbs:12569A 3478BC; ^ of Bases=441; 3-sum-decomp, info: BTS (3478BC,12569A), of sizes ( 6 , 6 ) and ranks (4,4), respectively. By /3\7/l\9 get minor 2 3 12 14 1489 = W4 Summands: Left= G_s. Right = G—4 . A's: 159 26A; J^'s: 37B 48C; ®'s : 15926A 37B48C; The output for R 12 shows that this matroid is kBj-decomposable, with the summands being the polygon matroid of the augmented 7rg 3 graph and its dual. The next 3 results are from [5]. The summing operations mentioned in these results were defined in Section 5.5. Proposition 8.4.6. If M is regular, and M has a minor isomorphic to R[ 2 , then M has an exact 3-separation { X [ ,X 2 ) with |ATi|,|X 2 | > 4 (indeed, > Q). Proposition 8.4.7. Let M be a 3-connected regular matroid. Then either M is graphic or cographic or M has a minor isomorphic to one of R\q, R \ 2 - Proposition 8.4.8 (Main Result of [5]). Every regular matroid M may he constructed by means of 1-, 2- and 3-sums, starting with matroids each isomor phic to a minor of M and each either graphic or cographic or isomorphic to R iq. 189 Using these results in conduction with Proposition 8.4.4 we obtain the following. Proposition 8.4.9. Let M be a 3-connected regular matroid in Sd^. Then either M in graphic or cographic or M has a minor isomorphic to Rp)- Proof. This foUows from Propositions 8.4.7 and 8.4.4. B Proposition 8.4.10. Every regular matroid M of may be constructed by m.eans of 1-, 2- and 3-sums, starting with matroids each isom.orphic to a minor of M and each either graphic or cographic. Proof. This foUows from Propositions 8.4.8 and 8.4.4. B 8.5. Generators of ^ 3 From Chapter 7 it follows that the polygon matroids of fig, W 4 , /v"3 ^3 , / i 5 \e, the 3-prism (= (K^\e)*), Æ3 3 -f e, W5 and the octacube are all generators of i^3 , with the first 3 inactive and the last 5 active. By taking duals, we obtain 2 more generators. In addition to these generators, 6 more are found in Appendix A. These 6 non-regular generators are all active with the last 4 being self dual. Table A1.5 gives the distribution of these generators while they are Hsted in List A1.7. Our first result hsts all known generators of ^ 3 and notes that each of them is either a graph or a graft. Proposition 8.5.1. Each of the following 3-connected binary matroids is a gen erator of (1) The polygon matroids of W^,, W 4 , K 3 3 , K^\e, the 3-prism (= (K^\e)*), + e, VF5 and the octacube, with the first 3 inactive and the last 5 active. (2) The cographic matroids and (M (Lf3 _ 3 4 - e))* with the former inactive and the latter active. (3) The non-regular matroids N\ = Fy, N 2 — F j , A 3 , N\q, N\^g and A 1 5 9 , all active. 190 Also, every ^-connected generator listed above is either a graph or a graft. Proof. The assertion that the 3 kinds of matroids are generators for ^ 3 follows from Chapter 7, dualization and Appendix A, respectively. We complete the proof by showing that every matroid in (2) and (3) is a graft. In order to prove that a binary matroid M is a graft, we first obtain either Rep(M) or Rep(M). Then we perform row reductions on Rep(M) until we obtain a matrix which has only 1 column that has more than 2 non-zero entries. We refer to this column as the graft column. Finally, we augment this matrix with a row wliich is equal to the sum of all the other rows, so that we can draw the graft corresponding to the matrix. We denote this augmented matrix by Graft(M). Suppose M = (M(7v3^3))*. Figure 8 . 8 gives the graph of A 3 3 with edges labeled by binary vectors coded as integers. 4 16 21 F ig u re 8 .8 . The graph R 3 3 with edges labeled by vectors Then, Rep(M(JF 3 3 )) and Graft(M) are as follows. /I 2 4 8 16 11 13 19 2 1 X 1 0 0 0 0 1 1 1 1 Rep(M(J\3,3)) 0 1 0 0 0 1 0 1 0 0 0 1 0 0 0 1 0 1 0 0 0 1 0 1 1 0 0 Vo 0 0 0 1 0 0 1 1 / /I 1 0 1 0 1 0 0 0 \ 1 0 1 1 0 0 1 0 0 0& T2) G raft(M ) = 1 1 0 0 1 0 0 1 0 1 0 1 0 1 0 0 0 1 \ 0 0 0 0 0 1 1 1 1 / 191 The graft column is the 1®*“, and the graft is illustrated by Figure 8.9 (a). (a) (M(K3.3))' Figure 8.9. The matroids g))* and + e))* as grafts Suppose M = 3 + 3))*. Then by utilizing Figure 8.8 and equation 8-11 we obtain the following. /I 2 4 8 16 11 13 19 21 3 \ 1 0 0 0 0 1 1 1 1 1 0 1 0 0 0 1 0 1 0 1 (Ebl3) R ep (M (/f 3 ^ 3 d- e)) = 0 0 1 0 0 0 1 0 1 0 0 0 0 1 0 1 1 0 0 0 Vo 0 0 0 1 0 0 1 1 0 / / I 1 0 1 0 1 0 0 0 0 \ 1 0 1 1 0 0 1 0 0 0 (8-14) R ep(M ) = 1 I 0 0 1 0 0 1 0 0 1 0 1 0 1 0 0 0 1 0 \1 1 0 0 0 0 0 0 0 1 / By performing i ? 5 -t- i ? 3 -4- i ? 3 and augmenting the resulting matrix with the additional row we obtain / I 1 0 1 0 1 0 0 0 0 \ 1 0 1 1 0 0 1 0 0 0 0 0 0 0 1 0 0 1 0 1 (8-15) G raft(M ) = 1 0 1 0 1 0 0 0 1 0 1 1 0 0 0 0 0 0 0 1 \ 0 0 0 0 0 1 1 1 1 0 / The graft column is the 1®\ and the graft is illustrated in Figure 8.8 (b). For F j we have / I 0 0 1 1 0 1 \ 0 1 0 1 0 1 1 (8-16) Graft(Fy) 0 0 1 0 1 1 1 \1 1 1 0 0 0 1 / The graft column is the last, and the graft is illustrated in Figure 8.10 (a). 192 Figure 8.10. The matroids Fj and Fj as grafts For F j we obtain 1 1 0 1 0 0 0 ' 1 0 1 0 1 0 0 R e p ( F f ) = (&T7) 0 1 1 0 0 1 0 1 1 1 0 0 0 1, By performing + R 2 + R i R\ and augmenting the resulting matrix with the additional row we obtain /O 0 0 1 1 1 0 \ 1 0 1 0 1 0 0 (8 -18 ) Graft(Ff) = 0 1 1 0 0 1 0 1 1 1 0 0 0 1 Vo 0 1 1 0 0 1 / he graft column is the and the graft is illustrated in Next Suppose M = N^. T hen /I 2 4 8 11 13 14 3 \ 1 0 0 0 1 1 0 1 (&T9 ) R ep(M ) — 0 1 0 0 1 0 1 1 0 0 1 0 0 1 1 0 Vo 0 0 1 1 1 1 oJ By performing R 3 + Ro + R\ Rl and augmenting the resulting matrix with the additional row we obtain /I 1 1 0 0 0 0 0 \ 0 1 0 0 1 0 1 1 (8-20) G raft (N 3 ) = 0 0 1 0 0 1 1 0 0 0 0 1 1 1 1 0 V i 0 0 1 0 0 1 1 / The graft column is the and the graft is illustrated in Figure 8.11 (a). 193 F ig u re 8 . 1 1 . The matroids and 7V%g as grafts For M ^ N iq we have /I 2 4 8 16 27 13 2 2 7 9 \ 1 0 0 0 0 1 1 0 1 1 (8-21) Rep(M) = 0 1 0 0 0 1 0 1 1 0 0 0 1 0 0 0 1 1 1 0 0 0 0 1 0 1 1 0 0 1 \ 0 0 0 0 1 1 0 1 0 0 / By performing R 4 + R\ -4- i?i and R 2 + R^ — -R3 , and augmenting the resulting matrix with the additional row we obtain / I 0 0 1 0 0 0 0 1 0 1 0 0 0 1 0 1 1 0110011000 (8-22) Graft(A^ig) = 0001011001 0 0 0 0 1 1 0 1 0 0 Vl 0 1 0 1 0 0 0 0 1/ The graft column is the 6 *"^% and the graft is illustrated in Figure 8.11 (b). Next Suppose M = Then /I 2 4 8 16 32 59 13 30 3 37 24 X 1 0 0 0 0 0 1 1 0 1 1 0 0 1 0 0 0 0 1 0 1 1 0 0 (8-23) Rep(M) = 0 0 1 0 0 0 0 1 1 0 1 0 0 0 0 1 0 0 1 1 1 0 0 1 0 0 0 0 1 0 1 0 1 0 0 1 Vo 0 0 0 0 1 1 0 0 0 1 0 / By performing R\ + R 2 + i ? 3 ^ i ? 3 and R^ + R 4 J?4 , and augmenting the resulting matrix with the additional row we obtain / I 0 0 0 0 0 1 1 0 1 1 0 \ 010000101100 111000000000 (8-24) G raft(N i5g) = 000110010000 000010101001 000001100010 \0 0110100000 1/ 194 The graft column is the 7*"^', and the graft is illustrated in Figure 8.12 (a). F ig u re 8 . 1 2 . The matroids and as grafts Finally, suppose M = A^isg. Then /I 2 4 8 16 32 59 13 30 3 37 29 \ 1 0 0 0 0 0 1 1 0 1 1 1 0 1 0 0 0 0 1 0 1 1 0 0 (8-25) Rep(M) = 0 0 1 0 0 0 0 1 1 0 1 1 0 0 0 1 0 0 1 1 1 0 0 1 0 0 0 0 1 0 1 0 1 0 0 1 \ 0 0 0 0 0 1 1 0 0 0 1 0 y By perform ing first Rq + R 4 + R^ -4- R^ and then R^ + R 4 —> R 4 , and augmenting the resulting matrix with the additional row we obtain /I 0 0 0 0 0 1 1 0 1 1 1 \ 0 1 0 0 0 0 1 0 1 1 0 0 0 0 1 1 0 1 0 0 0 0 0 0 (8-26) Graft (A^’i 59) = 0 0 0 1 1 0 0 1 0 0 0 0 0 0 0 0 1 0 1 0 1 0 0 1 0 0 0 0 0 1 1 0 0 0 1 0 V l 1 1 0 0 0 0 0 0 0 0 0 / 7th and the graft is illustrated in Figure 8.12 (b) Figure 8.13 illustrates the sixteen 3-connected generators of ^ 3 , given above. Table A1.6 gives the decomposition-type distribution for non-regular matroids in .â?3 , that have at most 12 elements. It has been verified that all such ma troids that have a big 3-separation or a 4-circuit/bond 3-separation are decom posable. There are only 5 matroids, wliich have a big 3-separation, that are of decomposition-type jVg but not of decomposition-type W4 . These matroids are hsted in List 9a. It is also shown that there is only 1 matroid, which has a big 195 J^15 (l'-'3,3 + e) octacube Figure 8.13. Sixteen 3-connected generators of as graphs or grafts 3-separation, that is of decomposition-type Æ 4 but not of decomposition-type or N^. This matroid is hsted in List 9b. List 9c gives 16 matroids that are of decomposition-type Fj, but not of decomposition-type IL 4 , 7V3 or N 4 . Each 196 of these matroids has a 4-circuit/bond 3-separation but no big 3-separation. On the strength of the evidence presented by the output and our work in Chapter 7, we make the following conjecture. Conjecture 8.5.1. Suppose M E and M is 3-connected. Then, if M M iW ^) and M has a big 3-separation then M is X^-decomposable, where Xg = W4 or Xg = or Xg = X 4 . Suppose M E Then Appendix A verifies this conjecture for non-regular matroids that have at most 12 elements. If a matroid M has more than 12 elements then by the central-edge-lemma M has a big 3-separation. Hence, for non-regular matroids we have the following consequence. Consequence 8.5.1. Suppose M E ^ 3. Suppose also that M is non-regular, 3-connected and has a big 3-separation. Then M is Xg-decomposable, where Xg = W4 or Xg = X 3 or Xg = N 4 . Proof. This follows from Appendix A, the central-edge-lemma and the vahdity of Conjecture 8.5.1. B We conclude with the following characterization of a generator. Consequence 8.5.2. Suppose M E ^ 3 and M is 3-connected. Then, M is a generator of if and only if M = M{W^), or M has no big 3-separation, or M has no A-circuit/bond 3-separation. Proof. First, suppose M is a generator of ^ 3 . Without loss of generahty, suppose M 5É M{W^). Then M has no big 3-separation, because otherwise by Conjecture 8.5.1. it foUows that M is decomposable. Therefore, by the central-edge-lemma, M has at most 12 elements. In order to prove that M has no 4-circuit/bond 3- separation, we first note that this is true if M is regular, by Proposition 8.4.2. So, suppose M is non-regular. Then M has no 4-circuit/bond 3-separation, because otherwise, by A ppendix A, it foUows th a t M is of decompo.sition-type Fy, which is a contradiction. 197 Conversely, if M = M{W^) then M is a generator, by Proposition 7.4.3. Also, if M has no big 3-separation or 4-circuit/bond 3-separation then M is a generator, by the definitions of the 5 summing operations. B 8 .6 . Obstacles of ^ 3 From Chapter 7 it follows that the polygon matroids of Ji5, Vg, the cube and the octahedron are all obstacles to S8 ^. By taking duals, we obtain 2 more obstacles. Also, from Proposition 8.4.4 it follows that i?io is an obstacle. In addition to these obstacles, 3 more are found in Appendix A. They are hsted in List A1.3. We hst all known obstacles to ^ 3 and prove that each of them is either a graph or a graft. Proposition 8.6.1. The following binary matroids are all obstacles to ^ 3 . (1) The polygon matroids of K^, kg, the cube and the octahedron. (2) The cographic matroids {M{K^))* and {M{Vg))*. (3) The regular matroid R iq. (4) The non-regular matroids N \i, A^i4 and A^23- Also, every obstacle listed above is either a graph or a graft. Proof. That the matroids in (1) and (2) are obstacles follow from Proposition 7.9.1 and duafization, respectively. Each of the matroids hsted in (3) and (4) consists of 10 elements and has no 4-structure. Therefore, each of these is an obstacle, by Proposition 6.5.2. We complete the proof by showing that the matroids hsted in (2), (3) and (4) are grafts. We prove this as in Proposition 8.5.1. Figure 8.14 gives the graphs of and Vg with the edges of each graph labeled by corresponding binary vectors that have been coded as integers. 198 1 2 12 127 30 / isX . / e o 10 \ 1 2 0 64 / 32 16 \ (a)A^ (a) Figure 8.14. The graphs of and Fg with edges labeled by vectors Suppose M = (M (/v 5 ))*. Then Rep(M(/v 5 )) and Rep(M) are as follows. /I 2 4 8 3 5 9 6 10 12\ 1 0 0 0 1 1 1 0 0 0 (8-27) Rep(M(#s)) 0 1 0 0 1 0 0 1 1 0 0 0 1 0 0 1 0 1 0 1 Vo 0 0 1 0 0 1 0 1 1 / /I 1 0 0 1 0 0 0 0 0 \ 1 0 1 0 0 1 0 0 0 0 1 0 0 1 0 0 1 0 0 0 Rep(M ) 0 1 1 0 0 0 0 1 0 0 0 1 0 1 0 0 0 0 1 0 Vo 0 1 1 0 0 0 0 0 1 ) By performing + R \ -)• Ri and Rq + R 2 ^ R 2 , and augmenting the resulting matrix with the additional row we obtain /O 0 0 0 1 0 1 0 1 0 \ 1 0 0 1 0 1 0 0 0 1 1 0 0 1 0 0 1 0 0 0 (&29) G raft(M ) = 0 1 1 0 0 0 0 1 0 0 0 1 0 1 0 0 0 0 1 0 0 0 1 1 0 0 0 0 0 1 \0 0 0 0 1 1 0 1 0 0/ The graft column is the 4*'^, and the graft is illustrated in Figure 8.15 (a). 199 Figure 8.15. The matroids {M{K^))* and {M{V^))* as grafts Next, suppose M = {M{V^))*. Then R ep (M (l^)) and R ep(M ) are as foUows. /I 2 4 8 16 32 64 15 30 60 120 127 \ 1 0 0 0 0 0 0 1 0 0 0 1 0 1 0 0 0 0 0 1 1 0 0 1 0 1 1 1 0 1 (8-30) Rep(M(1/%)) 0 0 0 0 0 1 0 0 0 1 0 0 0 1 1 1 1 1 0 0 0 0 1 0 0 0 1 1 1 1 0 0 0 0 0 1 0 0 0 1 1 1 Vo 0 0 0 0 0 1 0 0 0 1 1 / /I 1 1 1 0 0 0 1 0 0 0 0 \ 0 1 1 1 1 0 0 0 1 0 0 0 (61-31) R ep(M ) 0 0 1 1 1 1 0 0 0 1 0 0 0 0 0 1 1 1 1 0 0 0 1 0 Vl 1 1 1 1 1 1 0 0 0 0 1 / By performing i?2+ -^1 -Rl, -R3 + R2 R2, -R4+-R3 -4- R3, and R 4 -|-R 5 -> R5 in succession, and augmenting the resulting matrix with the additional row we obtain /I 0001001100 0\ 010001001100 0 0 1 0 0 0 1 0 0 1 1 0 (8-32) G raft(M ) 0 0 0 1 1 1 1 0 0 0 1 0 1 1 1 0 0 0 0 0 0 0 1 1 \0 001000100 1 1 / The graft column is the l l ’’*\ and the graft is illustrated in Figure 8.15 (b). For M = we have / I 2 4 8 16 19 25 28 14 7 \ 10 0 0 0 1 0 0 1 0 10 0 0 1 0 1 1 (8-33) R ep(M ) = 0 0 10 0 0 1 1 1 0 0 0 1 0 0 1 1 0 \0 0 0 0 1 1 1 0 0 / 200 By performing R 2 + R\ — R\ and i ? 4 + Ü 5 ^ i?5 , and augmenting the resulting matrix with the additional row we obtain /I 1 0 0 0 0 1 0 0 \ 0 1 0 0 0 1 0 0 1 0 0 1 0 0 0 0 1 1 (8-34) Graft(i?io) = 0 0 0 1 0 0 1 1 0 0 0 0 1 1 1 0 0 0 M 0 1 0 1 0 0 0 0 / The graft column is the 9*'*’, and the graft is illustrated in Figure 8.16 (a). Figure 8.16. The matroids i?io and # 2 3 as grafts For M = #23, we have / I 2 4 8 16 27 13 22 7 25 \ 10 0 0 0 1 1 1 0 10 0 0 1 1 0 Rep(M ) = (8-35) 0 0 10 0 0 1 0 0 0 0 1 0 1 0 0 1 \0 0 0 0 1 1 1 0 1 / By performing # 4 + —)■ i?i and i ? 2 + # 3 ->• #3 , and augmenting the resulting matrix with the additional row we obtain /I 0 0 1 0 0 0 0 1 0\ 0 1 0 0 0 1 0 1 1 0 0 1 1 0 0 1 1 0 0 0 (8-36) Graft (#2 3 ) = 0 0 0 1 0 1 1 0 0 1 0 0 0 0 1 1 0 1 0 1 \1 01010000 0/ The graft column is the 6*'*\ and the graft is illustrated in Figure 8.16 (b). Next, suppose M = # 1 1 . Then / I 2 4 8 11 13 14 3 5 9 \ 1 0 0 0 1 1 1 1 1 (Eh37) Rep(M ) 0 10 0 1 0 1 0 0 0 0 10 0 1 0 1 0 \0 0 0 1 1 1 0 0 1 / 201 By performing + R \ R i, and augmenting the resulting matrix with the additional row, we obtain /I 0 0 1 0 0 1 1 1 0\ 0 1 0 0 1 0 1 1 0 0 (8L38) Graft(A^ll ) = 0 0 1 0 0 1 1 0 1 0 0 0 0 1 1 1 1 0 0 1 Vl 1 1 0 0 0 0 0 0 1/ The graft column is the 7*"^, and the graft is illustrated in Figure 8.17 (a). (a) Figure 8.17. The matroids N \i and grafts Finally, suppose M = Then /I 1 0 1 1 0 0 0 0 0\ 1 0 1 1 0 1 0 0 0 0 0 1 1 1 0 0 1 0 0 0 (EL39) Rep(M ) 1 1 0 0 0 0 0 1 0 0 1 0 1 0 0 0 0 0 1 0 Vl 0 0 1 0 0 0 0 0 1/ + R\ — R l and fZg -|- R^, and resulting matrix with the additional row, we obtain /I 0 0 0 1 0 0 1 0 1\ 1 0 1 1 0 1 0 0 0 0 1 1 0 0 0 1 1 0 0 0 (8-40) Graft(A^jj) = 1 1 0 0 0 0 0 1 0 0 1 0 1 0 0 0 0 0 1 0 1 0 0 1 0 0 0 0 0 1 Vo 0 0 0 1 0 1 0 1 0/ The graft column is the 1® , and the graft is illustrated in Figure 8.17 (b). B Figure 8.18 illustrates all the obstacles found, as graphs or grafts. We conjec ture that the hst given in Proposition 8.6.1. is complete. 202 Conjecture 8 .6 . 1 . Any obstacle to ^ 3 is isomorphic to one of the matroids given in Proposition 8.6.1. Suppose M is an obstacle which makes the conjecture false. Then from Propo sition 7.9.5 it foUows th at M cannot be graphic or cographic. Also, from Propo sition 8.4.8 it foUows th at if M is regular then it must be a 3-sum of a graphic matroid and a cographic matroid. N\i Cube Octahedron R\o N 2 3 Figure 8.18. Ten obstacles to ^ 3 as graphs or grafts CHAPTER IX Non-binary Obstructions to Branch Decompositions 9.1. Introduction In this chapter we prove a result about uniform matroids, and a finiteness result. We begin with some terminology and notation. Definition. Suppose k is a non-negative integer. Let the lower ideal of all matroids M such that /3{M) < k he denoted by A matroid M is said to be an obstruction (or an obstacle) to a branch decomposition of width k if and only if M is an obstacle to N o ta tio n . Üâ3{k) = {[M\ : M is an obstruction to a branch decomposition of width k}. CO = ( J ûi^{k). (&T) k~0 ÛSd^ik) = {[M] : M is a binary matroid and [M] E ûS8 {k)}. OO ÜâS‘1 = (J ûsdoik). k= 0 We shall often refer to a matroid M as an obstruction to width k or an obstruction to branch-width k if [M] E ffB3{k). We shall also use the words “obstruction” and “obstacle” interchangeably. The question as to whether û^{k) is finite, for any given value of fc, is a very interesting one. Our first result answers it in the affirmative, for h < k <2. 203 204 Proposition 9.1.1. (1) û ^{0 ) = û'^2(0) = {[(^0 ,2 ], [(^2 ,2 ], [(^1 ,2 ], where G is a graph that consists of a loop and a link; (2) ^^(1) = (3) ^^(2) = {[M(7i:4)],[[;2,4]} ^ ^ 2 (2 ) = {[M(A"4)]}. Proof. (1) This follows from (2) of Proposition 5.2.3. (2) This follows from (2) of Proposition 5.3.3. (3) From (2) of Proposition 5.4.6 it follows that [Ad{K4 )] 6 ^.^(2), and that <^.^2 (2 ) = {[M(FL4 )]}. From (3) of Proposition 2.1.5 it follows that — 3- Since any proper minor of C/ 2 , 4 has branch-width less than 3 it follows that [C/2,4 ] G ^ ^ (2 ). Conversely, suppose [M] G ûâd{2). If Ad is non-binary then since C/ 2 , 4 m ust be a minor of M, it follows that M = C/ 2 ,4 . On the other hand, if A4 is binary then A4 ^ N4{4 We prove a useful necessary and sufficient condition for obstacles. Proposition 9.1.2. Suppose A4 is a matroid. Then [M] G if and only if (9-2) /3(M;.) < ;9(M) > /)(Mj.') Vz G g(M). Proof. Necessity follows from the definition of an obstacle and sufficiency is im plied by (4) of Proposition 2.1.3. B The next result looks at obstacles from the point of view of some farnihar matroid concepts and constructs. Proposition 9.1.3. Suppose [A4] G ûâS{k), where k > 0. Then: (1) [Ad*] G 1 then A4 is connected; (3) If k > 2 then Ad is simple. 205 Proof. (1) This follows from (2) of Proposition 2.1.3 and Proposition 9.1.2. (2) Assume M is not connected. Let C i,..., (7n be the components of M, where n > 2. Since Â: > 1, it follows th at (3{M) > 2. Hence, one of the components of M must have a non-loop circuit and tliis imphes that some component of M has at least 2 elements. Therefore, by (9) of Proposition 2.1.3, it follows that M has the same branch-width as one of its proper minors. This is a contradiction. (3) Assume M has a loop I or an element p that is parallel to another element. Let Ml be the proper minor of M obtained by deleting I or p, respectively. Since [M] G ûS8{k) where A; > 2, it follows that |5(M)| > 3. Then, by (5) of Proposition 2.1.3 it follows that f3{Mi) > 2. Once again, by (6) and (7) of Proposition 2.1.3, it follows that j3{M\) = j3{M), which is a contradiction. ■ The next result is about the existence of obstacles in general. Proposition 9.1.4. Suppose is a lower ideal of matroids and b > 0. If 3M G JT such that (3{M) = 6 4- 1, then 3N G such that N G û^{h). Proof. Consider the set M of all minors of M that have branch-width equal to 6 -f 1. Then M 0. Because M is finite, is also finite. The relation “is a minor of” is a partial order on M. Any minimal element of with respect to this partial order, is in Jd as well as Ûâ3(h). ■ Corollary. Vfc > 0, ÛSS2 [k) 0. Proof. By (1) of Proposition 2.3.1, the binary matroid that consists of all the vectors in the vector space (GF(2))^"‘^^ has branch-width equal to Â: -f- 1. The result now follows from Proposition 9.1.4. ■ It is known that there are polygon matroids of graphs (even planar graphs), that have any given branch-width. Hence, the above corollary may be appHed to these classes of matroids also. 9.2. Uniform matroid obstacles to branch-width We find the all uniform matroids that are obstacles to branch-width. 206 Proposition 9.2.1. The complete list of isomorphism classes of uniform ma troid obstacles to branch-width, is as follows, (1) [Uq,2]AU\,2]AU2,2] G û^{0); (2) e ^^(1); (3) Vg > 2, {[(7p,3g_2] : 9 < P < 2g - 2} Ç Proof. We first show that (1), (2) and (3) are true. From (1) and (2) of Propo sition 9.1.1 it follows that (1) and (2) are true. In order to prove (3), suppose q > 2 and q < p < 2q — 2. By Proposition 9.1.3, it follows that if [M] E Ûâ8 [q) then [M*] E &^[q). Therefore we may, without loss of generality, assume that q < P < |_(3g - 2)/2j. If X E S{Up^2q-2 ), then (9-3) (C/p,3,-2)L - C/p,3g-3 and {Up^ 2q-2)x = )^p-l,3g-3- From (5) of Proposition 2.1.5 it follows that P{Up,'iq- 2 ) = 1 +niin{|'(35-2)/3],p} = g + 1 (9-4) ^((/p,3g-3) = 1 + min{ [(3g - 3)/3] ,p} = q /)((/p-l,3g-3) = l + niin{[(3g-3)/3],p-l} = q. Therefore, from Proposition 9.1.2 it follows that Up^2q- 2 G û^{q). Conversely, suppose E Then, n > 2. Since every uniform matroid that consists of 2 elements is an obstacle, we suppose n > 3. Also, by the arguments given earher, we may without loss of generality let k < [n /2 j. As before, if x E 5'(17/. „), then (9-5) {Uk,n)'x - Uk,n-{ and (I7t,n)" - Because [Uk^n[ G from Proposition 9.1.2 it follows that P{Uk,n-\) < P{^k,n) > P{Uk-l,n-l)- Hence, min{ [(n — l)/3 ], k} < min{ [n/3], A;} > min{ [(n — l)/3 ], k — 1}. The left-side strict inequality imphes that n = l(mod 3) and that k > [n/3]. Now let n — 3q — 2. Then, since n > 3, it follows that q >2. Also, k > [n/3] — f(3g — 2)/3] = q. Therefore, q < k < [jî/2J = [(3g — 2)/2j and by considering the duals we have q < k < 2 q — 2 , as required. O 207 Uo,2 U\^2 U2,2 U l,2 (/2.4 (73,7 ( 4 ,7 (4,10 (4,10 (4,10 (75,13 (76,13 (77,13 (7s ,13 (76,16 (77,16 (7s ,16 (7g,16 (7lO,16 (77,19 (7g,19 (7g,19 (7 i O,19 (7i i ,19 (7]2,19 (7s,22 (7g,22 (7i0,22 Ui1,22 (7i2,22 (7i3,22 (7m,22 (7g,25 (7 io ,25 (7i i ,25 (7i 2,25 (7i 3,25 (7h ,25 (7i 5,25 (7i 6,25 (7io ,28 (7i i ,28 (7i 2,28 (7i 3,28 (7i 4,28 (7i 5,28 (7i 6,28 (7i 7,28 (7i 8,28 (7i 1,31 (7i 2,31 (7i 3,3 i (7i 4,3 i (7i 5,31 (7l6,31 (7l7,31 (7i 8,31 (7i 9,31 (720,31 (7i 2,34 (7i 3,34 (7h ,34 (7i 5,34 (7i 6,34 (7i 7,34 (7i 8,34 (7i 9,34 (720,34 (721,34 (722,34 Figure 9.1. Some uniform matroid obstacles to branch decomposition Proposition 9.2.1 guarantees the existence of fc — 1 non-binary obstacles to width Â:, Vfc > 2. Figure 9.1 illustrates the first few uniform matroid obstacles. In this figure the uniform matroids in the row are obstacles to branch-width & — 1 . 9.3. Obstacles with highest branch-width for a fixed rank To facihtate our discussion we introduce further notation. N otation. Suppose k and r are integers. Then let â'â3{k,r) = {[M] G ûâS{k) : p{M) = r}, where 0 < k < r Ûâ8 2 {k^r) = {[M] G Û3d2{k) : p{M) = r}, where 0 < fc < r — 1 208 From (3) of Proposition 2.1.3 and (2) of Proposition 2.3.1, it foUows that û^{k,r) = 0 if r < and = 0 if 1 < ?■ — 1 < respectively. In this section we are interested in û^{r,r), where r > 0. We have already seen that (9-8) û^{0,0) = {[î7o,2]}, ^ ^ (1 ,1 ) = {[^ 1 ,2 ]} and Û^{2,2) = {[Z/g,!]}- If [M] G ÛS8 {r,r), then p{M) = r and 0{M) = r + 1. Therefore, if r > 2, then by Proposition 2.3.1, it follows that M is not a binary matroid. We prove th at ÛS3{r,r) is finite by showing th at \S{M)\ is bounded by a function of r, if [Ad] G &38{r^r). Thereafter we improve the upper bound for |5(M )|. We begin by proving a necessary and sufficient condition for membership in û^{r,r). Proposition 9.3.1. Let Ad be a matroid with p{Ad) = r > 2. Then [Ad] G û^{r,r) if and only if S[Ad] is not the union of any 3 hyperplanes of Ad, and S{Ad'j.) is the union of 3 distinct hyperplanes of Ad, Vz G S{Ad). Proof. First suppose [Ad] G ÛSS[r,v). Then, from (2) of Proposition 2.1.2 it follows that 5(M) cannot be the union of 3 hyperplanes. Also, from (2) of Proposition 2.1.2 it follows that if z G S(M) then 5(M^) must be the union of 3 distinct hyperplanes of Ad. Conversely, suppose the conditions hold. Then, by (2) of Proposition 2.1.2 it foUows th at /9(M) — r -f 1. Let z G S{Ad). Then, once again, by (2) of Proposition 2.1.2 it follows that /l(M^) = v. From the hypotheses it foUows th at z cannot be a loop. Therefore, p(Ad'f) < r — 1. Hence, 0{Ad'J) < r. Therefore, [Ad] G û^{r,r). B We need the foUowing lemma. Lemma 9.3.1. Suppose t E Z. Then: (1) l f t > 0 , then ( < (3' - l)/2; (2) //"f > 2, f/ienf < (3' - l)/2; (3) Ift > 3, then ( < (3 ' - 7)/2. 209 Proof. We first observe that (1) is true for t — 0,1, and that (2) is true for t — 2. 3^ — 7 To complete the proof, we define / ; ]R — > IR by f{x) = — z Vz E R. Then /(3) = 7 > 0. Since ^ —- — 1 > 0, Vz > 3, the result follows. B Unless otherwise stated, let [M] E ûâS{r,r) for the remainder of this chap ter. We obtain a bound for the sizes of dependent subsets of M, in terms of hyperplanes. Lemma 9.3.2. Suppose [M] E ûS8 {r^r), where r > 2. Suppose also that X Ç S{M), with X ^ y{M). Then 3z E X and 3Hi, H 2 , E such that n (%\z)) < - 1 /or 1 < 2 < 3, and |%| < 1 + Z L l ^ (Z\z)|. Proof. Since % is dependent, 3C E V{M) such that C C X. Let z E C . Then z belongs to any flat of M, that contains X \z. By Proposition 9.3.1, it follows th at E Jlf(M) such that S{Ml) = Hj and X \ z 3 (9-9) |X| = 1 -b |X\z| <1 + 1: \Hi n (X\z)|. B i-l We obtain a bound for the sizes of subsets of M, in terms of their ranks. Proposition 9.3.2. Suppose [M] E Ûêd{r,r), where r > 2. Suppose also that 3P(A) _ I X Ç S{M). Then |X | < ------. Proof. Let t — p{X). We proceed by induction on t. Since r > 2, by (3) of Proposition 9.1.3, it follows that M is simple. Therefore, if i = 0 then |A| = 0 = (3® — l)/2, and if t — 1 then |X| = 1 = (3^ — l)/2. Hence, the result is valid for 7’ = 0 and r = 1. Assume that the result is true for t < p, where 0 < p < r — 1. Suppose t = j) + I. If A is independent then, by Lemma 9.3.1, it follows that |A| = p -|- 1 < (3^'*'^ — l)/2, as required. 210 Suppose X is dependent. Then, by Lemma 9.3.2, it follows that E % and 6 J^{M) such that p{Hi D (%\z)) < p{X) — 1 = p for 1 < i < 3, and 3 , o p _ 1 \ op+l _ 1 (9-10) |X| < 1 + E \Hi n (%\z)| < 1 + 3 ( ^ — j < ------ The result now follows by induction. B We can now prove that ûâS{r, r) is finite. Proposition 9.3.3. û'^[r,r) is finite, V integer r > 0. Proof. We have already seen that û^{r, r) is finite, for r = 0,1,2. Suppose r > 3. If [M] 6 û^[r,r) then by Proposition 9.3.2 it follows that |5'(M)| < (3'’ — l)/2. Since there are only a finite number of pairwise non-isomorphic matroids on a set of bounded size, ÛS8 {r, r) is finite. B We now attempt to improve the upper bound given by Proposition 9.3.2. This bound cannot be improved for r — 2 , because [J/2,4] G G^{2 , 2 ), and the bound given by Proposition 9.3.2 is 4. So suppose r > 3. We prove a lemma about hyperplanes. Lemma 9.3.3. Suppose [M] G ÛS3{r,r), where r > 3. If H \,H 2 G such that \Hi\ = \H 2 \ = (S'-! - l)/2 then n H 2 0. Proof. Assume otherwise. Without loss of generahty, suppose H\ H2 - Since p{H\) = ;• —1 > 2, by Lem m a9.3.1 it follows that p{H\) = r —1 < (3’’“ * —1)/2 = |7Ll|. Therefore, 3C G V{M) such that C Ç H\. Let z E C. Then z belongs to any fiat that contains H\\z. Also, since p{H\) = r — 1 > 2, it follows that 0. By Proposition 9.3.1, it follows that 3Aj, A 2 , A 3 G (M) such that = Uf=i and H \\z (f. Aj, for 1 < i < 3. Since H\ and A; are distinct hyper planes, p{{H\\z) n Aj) < p{H\ n Aj) < r — 2. Therefore, by Proposition 9.3.2, it follows that |(77i\z) n Aj| < (3'’“ ^ — l)/2. If |(77i\z) fl Afi < (3’’“ *"^ — l)/2, for some j such that 1 < ; < 3 then Ej=l 1(^1 V) A A,| < (3'-^ - 3)/2 = |77i\z|. This contradicts the fact that H \\z Ç U?_j Aj. Therefore, |(77i\z)nAj| = (3'’“ “ —1)/2 2 1 1 Vz such that 1 < i < 3. Also, since 1 = (3'’“ ^ — 3)/2, the sets {Hi\z) f l A, have to be pairwise disjoint. Since r > 3, it follows that {Hi\z)r\Ai ^ 0. Therefore, Ho ^ A{, for 1 < z < 3. Since H2 and A{ are distinct hyperplanes, p{H 2 f l Aj) < r — 2. Therefore, by Proposition 9.3.2, it follows that \H2 f l Aj| < (3'’“ ^ — l)/2. Hence we have that X )j-i \H2 f l Aj| < 3(3'’“^ — l)/2 < (3’’“ ^ — l)/2 — IH2 I, which contradicts the fact th at H2 Ç S{M )\z = Uj=i Aj. H Proposition 9.3.4. Suppose [M\ G û^{r^r), where r > 3. Then |.S(M )| < Proof. Let z G S{M). Then, by Proposition 9.3.1, it follows that 3H\, H21 H^ G JP[M) such that S{M'.) = Therefore, i5(M)| < 1 + Ej=i \Hi\- By Proposition 9.3.2, it follows that |iJj| < (3'’~^ — l)/2, for 1 < z < 3. Let n = |{z : |Lfj| = (3'”“^ — 1)/2}|. We argue by considering the cases corresponding to n < 1, rz = 2 and n = 3 separately. 2 \ 2 / 2 n = 2 : Without loss of generality, let |ifi| — |ffo| = — l)/2 and |jTg| < (3'’“ ^ — 3)/2. Then, by Lemma 9.3.3, it foUows that H\ fl H 2 ^ 0- Therefore, \Hi U H 2 I < 3'-! - 2. Hence, |5 (M)| - 1 + |H i U H 2 I + IH3 I < 1 + 3^~l - 2 + 3'-^ - 3 3’’ - 5 2 2 rz = 3 : By Lem m a 9.3.3, it foUows that Lfj fl Hj ^ 0, for 1 < z < y < 3. Therefore, |5(M)1 < 1 “ 2 - ■ We prove a further lemma about hyperplanes. Lemma 9.3.4. Suppose [M] G Ûü8 {r,r), where r > 3. Suppose also that |5(M )| = (3'’— 5)/2. Then: (1) If X G S{M) then 3Hx 6 (M ) such that x ^ Hx and \Hx\ = (3'’“ ^ —1)/2; (2) 3K[, K 2, G such that \Ki\ = (3'’~^ — l)/2 for 1 < z < 3, with Kf n Kj 0 for 1 < z < y < 3, and Ki 11 K 2 K\ fl Kz- Moreover, 2 1 2 Proof. (1) By Proposition 9.3.1, it follows that 3L\, L 2 , 6 such that S{M'x) — U ;_i Li. Therefore, |5(M)| < 1 + l^zi- It suffices to show that \Lj\ = —1)/2, for some j such that 1 < j < 3, for then we may let Hx = Lj. From Proposition 9.3.2, it follows that |L;| < (3'’~^ — l)/2 for 1 < i < 3. Assume \Li\ < (3'-l - 3)/2, for 1 < i < 3. Then \S{M)\ < 1 + Ef=i 14'I < 1 + 3 ^ ----- ^ = ------, which is a contradiction. \ 2 J 2 (2) Let a g S{M) and define K\ — Ha- Since r > 1, we have that Ki ^ 0. Let b G A'l and define ATg = Lfj. Then K\ ^ Ag. Since |Ai| = |Ag| = (3’’“ ^ — l)/2, by Lemma9.3.3, it follows that AiflA'g ^ 0. Let c G A^nAg and define A 3 = Ac. Then, Aj, Ag and A 3 are pairwise distinct. Also, since IA 3 I = (3'’~^ — l)/2, by Lemma 9.3.3, it follows that A% fl A 3 7 ^ 0 ^ Ag fl A 3 . Since c G A% n Ag and c ^ A 3 , it follows that K\ fl Ag 7^ A% H A 3 . From what has already been proved, we have that U = E IA-,1 - E |A'.-nA^|+ n/c, i—\ 2=1 2=1 o)—1 (9-11) =3(- ^ ' 1<2 So it suffices to show that n — Z!i<2 Since A% fl Ag 7 ^ K\ D A 3 , it follows that K\ fl Ag g flf=i A,. Therefore, |A j n Ag| > 1 + nf=i A;. Therefore n > 3. B We make a final improvement to the upper bound for |5(M )|. Proposition 9.3.5. Suppose [M] G û^{r,r), where r > 3. Then \S{M)\ < 3 ^ - 7 Proof. By Proposition 9.3.4, we have that |5(M )| < (S'" — 5)/2. Assume that IS'(M)! = (3'” — 5)/2. Let Ai, Ag and A 3 be the hyperplanes given by Lemma 213 9.3.4 and U = Then, \U\ < (3'’ - 9)/2. Let z G S{M)\U. Then, by Proposition 9.3.1, it follows that 3iJi, i?2)-^3 G (M) such that S{M'x) = We show that {K\, K 2 , K^} — { iîi, i? 2 , iîs}. Without loss of generality, as sum e th at K\ ^ Hi, for 1 < i < 3. Then p{K\ fl Hi) < r — 2 and, by Proposition 9.3.2, it foUows th a t |A'i fl Hi\ < (3'’~“ — l)/2. Since A'] = Uf=i(-bTi fl H{), it foUows th at (9 - 12) 3'"-^ - 1 Sr-2 _ 2 3'"7*—1 = m< 2=1 2=1 ^ which is a contradiction. Therefore, |5'(M)| = 14- lUf=i Hi\ = 1 4- |(7| < 1 4- (3'^ — 9)/2 = (3^ — 7)/2, which is also a contradiction. H In order to see that the bound given in Proposition 9.3.5 cannot be improved S'* - c to one of the form —-— where c > 7, we construct a matroid N of rank 3 such th at S{N) = {a,b,c,d,e,f,g,h,i,j} and (9-13) ■^{N) = {{a,b,c,d},{d,e,f,g},{g,h,b,i},{i,c,e,j],{j,f,h,a}} (See Figure 9.2). a Figure 9.2. A geometric representation of the matroid N 214 From the figure it can be seen that this rank 3 matroid N satisfies the condi- 3 3 _ 7 tions of Proposition 9.3.1. Hence, [A^] 6 ^.^(3,3). Also |5'(M)| = 10 = —-— . The next result gives the smallest number of elements a matroid M must have in order that [M] E ûâS{r,r), where r > 2. It also characterizes such matroids. Proposition 9.3.6. Suppose [M] G where r >2. Then: (1) |5'(M )|>3r-2,- (2) \S{M)\ = 3r — 2 if and only if M = I/r,3 r - 2 - Proof. (1) Assume |5(M )| < 3r — 3. Then 3Ai, A2 , A 3 6 S{M) such that |Aj| < r — 1 for 1 < I < 3, and S{M) = Uf=i ^i- Since p(Aj) < \Af\ < r — 1 , it follows th a t S[M) is the union of 3 hyperplanes of M. This contradicts Proposition 9.3.1. (2) First, suppose that M = (7r,3r-2- By Proposition 9.3.1, it follows that [M] G û^{r,r). Also, |5(Af)| — |5'(17r,3r-2)l = 3r — 2, as required. Conversely, suppose |5'(M)| = 3r —2. Assume M % Urfir-2- T hen 3D Ç S{M) such that |D| — r and D ^ y{M). Then |5'(M)\D| = 2v — 2. Therefore, 3X ,y Ç 5(M) such that |AT| = |F| = r - 1 and S{M)\D = X UY. Then S{M) = D \J X [JY and each of D, X and Y is contained in a hyperplane of M. This contradicts Proposition 9.3.1. ■ A P P E N D IX A Program Output This appendix consists of the output from a computer program which is briefly described in Appendix C. The major portion of this output is presented in the form of hsts and tables. The main purpose of this output is to provide data that is essential for un derstanding the behaviour of non-regular matroids in the ideal ^ 3 . This data is in fact provided in the lists A.l and A.2. The other hsts and tables serve to group the non-regular matroids that have a particular property and show how such matroids are distributed. AU hsts consist of data about matroids. Every matroid that appears in this output is binary and 3-connected. Also, each of these matroids is either graphic or cographic or non-regular. A matroid that is graphic or cographic is either an active generator or a summand, for 3-summing in ^ 3 . The hst A.l presents data about matroids that are either graphic or cographic, while all other hsts consist of data about non-regular matroids. In our explanations of matroid-data, we state data-items in the order in wliich they appear in the output. Integers in the output that are in bold type denote binary vectors that have have been encoded by the process explained in Section 1.6. This encoding of vec tors, is one of the space-saving devices that is employed in our effort to present the output in a reasonable amount of space. We now introduce a hst of abbrevi ations that are used. 215 216 Abbreviation I te m Bits' standard complement C-psorbs column pseudo-orbits R-psorbs row pseudo-orbits Exts extensions Cys cycles Cocys CO cycles Non-loc. non-localizable P rank \s\ # of elements A triangle A triad YA wye-delta 1X 1 4-circuit/bond © 6-structure ACT active 3-sum generator DOR inactive 3-sum generator DEC 3-sum decomposable matroid M ’s m atroids info information BTS big 3-separation FCBTS 4-circuit/bond 3-separation P m ean (T standard deviation The last 2 abbreviations are used in Table A.7 and Table A.8, wliich provide statistical information about the distribution of the number of bases and the number of column pseudo-orbits in a matroid. In these 2 tables /j denotes the fraction of matroids for wliich the quantity in question hes in the interval [/i — i(T, ft + icr], for 1 < i < 3. An emperical rule in the theory of statistics states that for a normal (bell-shaped) distribution, f\ > .68, / 2 > .95 and /g > .997. From our tables we can observe the extent to which the above distributions satisfy this emperical rule. 217 Suppose M is a binary matroid that appears in the output. Then M is identi fied by a symbol. If M is graphic or cograpliic then it is assigned a symbol of the form G{ where i < 0 , and if M is non-regular then it is assigned a symbol of the form Nj where j > 0. The grapliic matroids G'_i 2 , G _ 7 , G-q and G_i, which are self-dual, correspond to W4 , the augmented 3-prism, and the octacube, respectively. The matroid G _n is isomorphic to 3 ))*, while the grapliic m atroids G'_g, G _ 5 and G_g correspond to the graphs obtained by deleting an edge from A' 5 , the augmented 3 graph and the graph obtained by deleting an edge from the octahedron, respectively. The matroids G_io, G_g, G _ 4 and G _ 2 which are generated as duals, correspond to the 3-prism, G I 5 and the graph obtained by contracting an edge of the cube. The matroids G_?, G_g and G _ 2 are 3-sum decomposable, but they are included in List A.l because they appear as summands for 3-summing. The graphs corresponding to G_ 7 , G _ 3 and G _ 2 are illustrated in Figure A.l (a), {b) and (c), respectively. (a) Figure A .l. The graphs corresponding to the non-generators in List A.l The Fano matroid F-j is given as N\. The matroid M was obtained either from the input to the program, or by a single element extension or dualization of another 3-connected matroid of the output. In this output, the matroids G_% 2 , G _ ii, G _ 9 G _ 7 , G _ 6 , G _ 5 , G _ 3 , G_i and were obtained from an input, while all the other matroids were obtained otherwise. We note that N\ was in fact obtained internally, via an initialization. We refer to a matroid that is from the input as an input-matroid. If M was not obtained from the input then the matroid from which M was obtained is called the parent of M . On the other hand, if M was obtained from the input then M has no parent. Also, the matroid M 218 may be generated either as a primal matroid or as a dual matroid, by the program. We refer to M as primally generated or dually generated if M was generated as a primal or as a dual, respectively. As far as the ground set S{M) is concerned, we assume that it contains the standard basis {1,2,4,... }. We refer to the set of elements of S{M) that are not in this standard basis as a standard complement in S{M). In the output for M we only give a standard complement in 5(M ), because S{M) itself can be constructed from such a standard complement. In our presentation the order in which the elements of the standard complement is Hsted, is important. This is because we wish to treat S{M) as an ordered set for notational purposes. We do so as follows. We first assume that the elements of the standard basis, in the order previously given, occur before the elements of the standard complement. Next we assume that the elements of the standard complement occur in the order in which they are hsted in the output. By treating S{M) as an ordered set we are able to refer to a specific element of M, by its position in the ordered set. In order to save space, we denote the position of an element of M by the hexadecimal digit corresponding to that position. So the first fifteen positions are denoted by 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E and F, respectively. Suppose X Ç S{M). Then any string of hexadecimal digits that corresponds to the positions of the elements of X is called a position string of X. Although X usually has more than one position string, X itself is uniquely determined by any of its position strings. We use position strings to hst many of the significant subsets of S{M). The data for M is given by several hnes, with such data given by more than one fine only in the hsts A.l and A.2, and only if M is primally generated. If M is an input matroid or a weU-known matroid then in the hsts A.l and A.2, the first hne of data is centered and consists of the symbol for M which is foUowed by the symbol = and the usual symbol for M. In the description given below, it is important to note that items hsted for the hne of data for M, appear on the {n + 1)*'*' hne of data if M is an input-matroid, and on the hne of data otherwise. We begin by hsting the items, on the first hne of data for M. (1) The symbol for M followed by a colon unless M is an input-matroid; (2) The ordered pair {p{M), |5'(M )|); 219 (3) The character ‘P ’ or ‘D’ depending on whether M was primally or dually generated; (4) Tills item which appears only if M has a parent, consists of the string ‘Ref:’ followed by the symbol of M's parent; (5) This item appears only if M has single element extensions that appear in the list A.2. It consists of the string ‘Exts=’ followed by the number of single element extensions of M that appear in A.2; (6) This item which appears only if M is primally generated, consists of the string ‘Dual=’ followed by the symbol of M's dual. If M is strictly self dual then that symbol has an asterisk as superscript; (7) The string ‘Elts':’ followed by the elements of a standard complement in g (M ); (8) The string '/) —’ followed by the value of f3{M); (9) This item which appears only if (S{M) = 3 consists of one of the strings ‘ACT’, ‘DOR’ or ‘DEC’, as explained in the hst of abbreviations. There are essentially 2 items that are given on the second hne of data for M. These are the number of cocycles of M and the number of cycles of M of different sizes. Because M and M* are both 3-connected it foUows th a t every cocycle and cycle of M has at least 3 elements. Also every such co cycle and cycle may have no more than |5'(M)| elements. The 2 items on the second row are the foUowing. (1) the string of Cocys:’ foUowed by an array that consists of \S{M)\ — 2 non-negative integers. The entry of this array is equcd to the number of CO cycles of M that consists of t -|- 2 elements. (2) the string of Cys:’ foUowed by an array that consists of |5(M)| — 2 non-negative integers. The entry of this array is equal to the number of cycles of M that consists of f -f 2 elements. The third Une of d ata for M gives information about the pseudo-orbits and the bases of M. A pseudo-orbit consists of a union of isomorphism-orbits. The data-item s are as foUows. (1) The string of R-psorbs=’, foUowed by the number of row pseudo-orbits of the row space of M; 2 2 0 (2) This item depends on the number of column pseudo-orbits of M. If the number of column pseudo-orbits of M is equal to 1 or the number of ele m ents of M then the item consists of the string of C-psorbs=’, followed by the number of column pseudo-orbits of M. Otherwise this item consists of the string ‘C-psorbs;’ followed by the column pseudo-orbits of M, given as position strings; (3) The string ‘ 7^ of Bases=’, followed by the number of bases of M. If P{M) = 3 an d M is 3-sum-decomposable then the output of the next few hnes depends on the decomposition-type of M. If M is of decomposition-type W4 , A^3 or iV4 then the decomposition information is given in 3 hnes. The items on these 3 hnes are as foUows: (1) The string ‘Decomposition info:’, foUowed by a colon; (2 ) The string ‘BTS’, wliich refers to a big 3-separation (X,Y); (3) The position strings of X and Y given as an ordered pair; (4) The phrase ‘of sizes’ foUowed by (|%|, |y|); (5) The phrase ‘and ranks’ foUowed by {p{X),p{Y)),, foUowed by the word ‘respectively.’; (6 ) Information about the construction of an 8 -element minor of M th a t is isomorphic to IT 4 , or #4 , is presented next. The word ‘By’ is foUowed by minoring operations of deletion and contraction each of which is in turn foUowed by the position of the element that is deleted or contracted, respectively. Deletion and contraction are denoted by ‘\ ’ and ‘/ ’, respec tively. After minoring, 4 elements of X as weU as 4 elements of Y rem ain to form an 8 -element minor. The minoring operations are foUowed by the phrase ‘get minor’, which in turn is foUowed by 8 binary vectors. These vectors are foUowed by one of the strings ‘= IT 4 ’, or ‘= #4 ’. (7) The next row begins with the word ‘Summands’ which is foUowed by a colon; (8 ) Let Ml and Mr be the left and right summands of M as given by this big 3-separation and minoring. Then the last two items on tliis row consists of ‘Left= M ;’ and ‘Right= Mr \ respectively. 221 If M is of decomposition-type F-j then the decomposition information is given in 2 hnes. The items on these 2 hnes are as follows: (1) The string ‘Decomposition info:’, foUowed by a colon; (2) The string ‘FCBTS’, which refers to a 3-separation (X, T), where X is a 4-circuit/bond of M; (3) The position strings of X and Y given as an ordered pair; (4) The next row begins with the word ‘Summands’ which is foUowed by a colon; (5) Let Ml and Mr be the left and right summands of M as given by this FCBTS, and the minoring that foUows. Since either M/ = Fj or M/ = Fj , the next item on this row is either ‘Left= Fj' or ‘Left= Fy’, respectively. This string is foUowed by the word ‘by’, which is foUowed by the minoring operations that gave rise to M/. The next item on this row is ‘Rights M r’, which is foUowed by the word ‘by’ which is foUowed by the minoring operation that gave rise to Mr- The rest of the data for M consist of a hsting of aU the smaU structures of M, with each smaU structure being given by its position string. The strings ‘A's:’, ‘Y A's:\ ‘xi' s:’ and ‘(s)'a:’ are foUowed by the sets of triangles, triads, wye-deltas, 4-circuit/bonds and 6 -structures of M, respectively. E a ‘-’ sign is written in front of a position string of a 3-structure then that 3-structure is non- locaUzable. Information about the small structures may be used to construct a branch decomposition for M. List A.2 gives aU non-regular 3-connected matroids, with at most 12 elements that are in ^ 3 or are obstacles to ^ 3 , up to isomorphism. AU other hsts of matroids except List A.l, are essentiaUy subUsts of List A.2. We note that matroids of branch-width 4 that are not obstacles to . ^ 3 do not occur in any of the hsts. In List A . 2 aU the matroids that have the same rank and the same number of elements are hsted in a contiguous block, except for page-breaks. Any two different blocks are separated by a few blank hnes. The contents of the different Usts and tables of this output are self explanatory. In each table the number of matroids that have certciin properties or contain certain structures are given, by rank and number of elements of such matroids. If for some rank and number of elements, there is no matroid that has any of the 222 relevant features then there is no row in the table that corresponds to that rank and that number of elements. The reader is encouraged to verify the accuracy of the program output given in this appendix. To this end, we have provided an addition table for binary vectors, in Appendix B. Although this table only gives binary vector sums ‘from’ 0 + 0 ‘to’ 64 + 64 we hope that it wiU be of some help. 223 List A .l. Polygon matroids of 3-connected graphic/cographic summands G_12 = (4, 8 ) P Dual=G_i2 Bits': 3 5 10 12 /3 = 3 ACT # of Cocys: (4,5,4,2,0,0);#ofCys: (4,5,4,2,0,0); # of R-psorbs=5; C-psorbs:1234 5678; # of Bases—45; A's: 125 136 247 348; A's: 156 257 368 478; YA 's: 1562 1653 2571 2754 3681 3864 4782 4873; ®'s : 125348 136247 156478 257368; G _ ii = (M(AT3,3))* (4, 9) P Dual=G_io Bits': 3 7 10 12 13 /? = 3 ACT # of Cocys: (0,9,0,6,0,0,0);#ofCys: (6,9,9,6,0,0,1); # of R-psorbs=2 ;# of C-psorbs=l; # of Bases—81; A's: 125 189 247 348 356 679 ; ®'s : 125348 125679 189247 189356 247356 348679 ; G_io: (5, 9) D Ref:G_ii Bits': 19 7 26 28 /? = 3 ACT G _ 9 = M{J<5 - edge) (4, 9) P D u al= G _ 8 Bits': 3 9 10 12 6 ^ = 3 ACT # of Cocys: (2,3,6,4,0,0,0);#ofCys: (7,9,6,6,3,0,0); # of R-psorbs=5; C-psorbs:135689 247; # of Bases=75; A's: 125 146 239 247 348 567 789; /^'s: 156 389; Y A's: 1562 1654 6517 3894 3982 9837; ®'s : 156389; G_g: (5, 9) D Ref:G _ 9 Bits': 3 21 24 14 ^ = 3 ACT G _ 7 = M(three-prism 4 - edge) (5,10) P Dual=G_ 7 Bits': 3 6 12 19 25 ^ = 3 DBC # of Cocys: (4,6,8,8,4,1,0,0); # of Cys: (4,6,8,8,4,1,0,0); # of R-psorbs=19; C-psorbs: 1479 2 36 58 A; # of Bases=130; Decomposition info: BTS (3478A,12569), of sizes (5,5) and ranks (4,3), respectively. By / A \2 get minor 2436 12 8 13 5 = W4 Summands: Left= G_g. Right = G_ 9 _ A's: 126 237 348 569; /^'s: 378 48A 59A 156; Y A's: 3782 3874 48A3 59A6 1652 6519; 224 G _ 6 = M (W s) (5.10) P Dual-G_c Bits': 3 6 12 24 17 /3 = 3 ACT # of Cocys: (5,5,6,10,5,0,0,0); # of Cys: (5,5,6,10,5,0,0,0); # of R-psorbs=7; C-psorbs:12345 6789A; # of Bases= 1 2 1 ; A's: 126 15A 237 348 459; ^s: 16A 267 378 489 59A; Y A's: 16A2 1A65 2671 2763 3782 3874 4893 4985 59A4 5A91; G _ 5 = M {K \s + edge) (5.10) P Dual=G_ 4 Bits': 3 11 14 19 22 /? = 3 ACT # of Cocys: (4,5,9,10,2,0,1,0);# of Cys: (3,9,6,6,7,0,0,0); # of R-psorbs=ll; C-psorbs: 124579 38A 6 ; # of Bases=135; A's: 126 467 569; /^'s: -38A 478 59A 123; Y A's: 4786 59A6 1236; ©'s : 47859A 478123 59A123 ; G_ 4 : (5,10) D Ref:G _ 5 Bits': 11 31 20 6 24 ^ = 3 ACT G _ 3 = M(octaliedron — edge) (5.11) P Dual=G_ 2 Bits': 3 6 14 15 24 31 /? = 3 DBC # of Cocys: (2,4,8 , 8 , 6 ,3,0,0,0); # of Cys: (6,10,14,16,10,5,2,0,0); # of R-psorbs—13; C-psorbs:14789 25 36AB; # of Bases=224; Decomposition info: BTS (4589AB,12367), of sizes (6,5) and ranks (4,3), respectively. By \4/ A \1 get minor 8 14 15 72436 = W4 Summands: Left= G_%. Riglit= G_g. A's: 126 189 237 45A 478 59B ; A'a: 5AB 236; Y A 's : 5AB4 5BA9 2367 2631; G_ 2 : (6,11) D Ref:G _ 3 Bits': 41 47 46 60 48 /? = 3 DBC G_i = M(octacube) (6.12) P Dual=G_i Bits': 3 12 15 22 55 56 ^3 = 3 ACT # of Cocys: (4,6,12,16,12,9,4, 0, 0,0); # of Cys: (4, 6,12,16,12,9,4, 0,0, 0); # of R-psorbs=15; C-psorbs: 1234BC 56A 789; # of Bases=392; A 's: 127 348 -789 9BC; /^'s: 6 BC 12A -56A 345; Y A's: CB69 12A7 3458; 225 List A.2. Non-regular 3-connected matroids with at most 12 elements (that are in ^ 3 or are obstacles to ^ 3 ) Ni (3, 7) P Dual=A2 Bits': 3 5 6 7 /3 = 3 ACT # of Cocys: (0,7,0,0,0); # of Cys: (7,7,0,0,1); # of R-psorbs=l;^ of C-psorbs=l; ^ of Bases=28; A's: 124 135 167 236 257 347 456; ex' s: 1237 1457 2345 2467 1346 3567 1256; N-2 : (4, 7) D Ref:Ai Exts=2 Bits': 1113 14 /3 = 3 ACT /V3 : (4, 8 ) P Ref:iV2 B xts = 2 D ual= # 3 B its': 11 13 14 3 ^ = 3 A C T # of Cocys: (3,7,4,0,1,0); # of Cys: ( 3 ,7 ,4 , 0 , 1 , 0 ); # of R-psorbs=5; C-psorbs:124567 3 8 ;# of Bases=48; A'a: 128 458 678; )|\'s: 367 123 345; Y A'a: 7638 1238 5438; M' a: 1245 4567 1267; A 4 = A G (3,2) (4, 8 ) P Ref:A 2 Dual=A| Bits': 11 13 14 7 /3 = 3 DBC # of Cocys: (0,14,0,0,0,1); # of Cys: (0,14,0,0,0,1); # of R-psorbs=2;# of C-psorbs=l; # of Bases=56; Decomposition info: FCBTS (1568,2347) Summands: Left= Fj by /2. Right = N\ by / I . m ' s: 1568 2347 2578 1346 3678 1245 4567 1238 1267 1357 1478 2356 2468 3458; A 5 : (4, 9) P Ref:A 3 Bxts=3 Dual=A7 B its': 11 13 14 3 5 ^ = 3 D EC # o f Cocys: (1,5,6,2,1,0,0); # o f Cys: (6,10,8,4,2,1,0); # of R-psorbs=7; C-psorbs:89 1 23 4567; # of Bases=80; Decomposition info: FCBTS (4567,12389) Summands: Left= Fj by \8\9. Right = G_j 2 by \4. A's: 128 139 458 469 579 678 ; 4 \'s: 123; Y A's: 1238 1329; tx's: 4567; Ac: (4, 9) P Ref:A 3 Dual=Ag Bits': 11 13 14 3 7 ^ = 3 DBC # of Cocys: (0,6, 8,0,0,1,0); # of Cys: (4,14,8,0,4,1,0); 226 # of R-psorbs=3; C-psorbs: 12345679 8 ; # of Bases=8 8 ; Decomposition info: FCBTS (3679,12458) Summands: Left= by \1\8. Rights by \3. A's: 128 389 458 678; tx's: 3679 1245 4567 1239 1267 3459; N^. (5, 9) D Ref: 7V5 E xts = 8 F its': 27 13 22 7 /) = 3 DEC iVg: (5, 9) D Ref:iV 6 Exts=2 Fits': 27 29 22 7 = 3 DEC # 9 : (4,10) P Ref:A^5 Exts=l Dual=A^i2 B its': 11 13 14 3 5 6 /? = 3 DEC # of Cocys: (1,1,6 , 6 ,1,0,0,0); # of Cys: (10,16,12,12,10,3,0,0); # of R-psorbs=5; C-psorbs:123 4567 89A; # of Bases=124; Decomposition info: BTS (4567A,12389), of sizes (5,5) and ranks (3,3), respectively. By \4\8 get minor 11 13 14 6 1 2 4 5 = W 4 Summands: Lefts 7V5 . Rights G_q. A's: 128 139 23A 458 469 47A 56A 579 678 89A; /|\'s: 123; Y A's: 1238 1329 321A; m 's: 4567; Niq: (4,10) P ReLiVs D ual=#i3 F its': 11 13 14 3 5 7 /3 = 3 DEC # of Cocys: (0,2,8,4,0,1,0,0); # ofCys: (8,18,16,8,8,5,0,0); # of R-psorbs—4; C-psorbs:89 1234567A; ^ of Bases=136; Decomposition info: BTS (45679,1238A), of sizes (5,5) and ranks (3,3), respectively. By \4\1 get minor 11 13 14 5 2 4 3 7 S W4 Summands: Lefts Rights N^. A's: 128 139 29A 38A 458 469 579 678; m ' s : 4567 123A; A^li: (4,10) P Ref:A^5 Dual^A^n Fits': 11 13 14 3 5 9 /? - 4 # of Cocys: (0, 3,6 ,4,2,0,0,0); # of Cys: (9,16,15,12, 7,3,1,0); # of R-psorbs=5; C-psorbs:15689A 237 4; ^ of Bases=131; A's: -128 -139 -14A -25A -36A -458 -469 -579 -678; Nx2 - (6,10) D Ref:Yg Fits': 27 45 54 7 /9 = 3 DEC Nu- (6,10) D ReLATp, Fits': 59 45 54 7 /? = 3 DEC Nu: (6,10) D ReLNn Fits': 59 13 22 39 /3 = 4 227 Nis: (4,11) P RefzATg Dual=iVi6 E lts': 11 13 14 3 5 6 7 /? = 3 DEC # of Cocys: (0,2,0,12,0,1,0,0,0); # of Cys: (13,26,24,24,26,13,0,0,1); # of R-psorbs=3; C-psorbs: 1234567B 89A; # ofBases=200; Decomposition info: BTS (1239AB,45678), of sizes (6,5) and ranks (3,3), respectively. By \1\9\4 get minor 2 4 6 7 11 13 14 3 = W4 Summands: LeftS Ag. Right = A 5 . A's: 128 139 lAB 23A 29B 38B 458 469 47A 56A 579 678 89A; tx's: 4567 123B; N iq: (7,11) D Ref:Ai5 Elts': 91 109 118 7 /3 = 3 DEC A 1 7 : (5,10) P Ref: Ay Exts=5 DuaI=A2 o E lts': 27 13 22 7 3 = 3 DEC # of Cocys: (2,10,8 ,4 , 6 ,1,0,0);# of Cys: (3,6,11,8,1,1,1,0); # of R-psorbs=12; C-psorbs:78 A 1239 45 6 ; # of Bases=144; Decomposition info: BTS (45678,1239A), of sizes (5,5) and ranks (4,3), respectively. By /4\1 get minor 8 11 5 14 2473 = W4 Summands: Left= G_g. Right= A 5 . A's: 12A 39A 678; /f\'s: 467 568; Y A's: 7648 8657; tx's: 1239; Aig: (5,10) P Ref:A? Exts = 6 Dual=Aig Elts': 27 13 22 7 5 /? = 3 DEC # of Cocys: (4,6 , 8 , 8 ,4,1,0,0); # of Cys: (4,6 , 8 , 8 ,4 ,1 ,0 ,0 ); # of R-psorbs=19; C-psorbs:123489 6 A 5 7; # of Bases=132; Decomposition info: BTS (4678A,12359), of sizes (5,5) and ranks (3,4), respectively. By \ A /5 get minor 8 11 13 6 1 2 4 7 = A 3 Summands: Left= A5 . Right= Ay. A's: 13A 29A 47A 678; 4^'s: 467 568 135 259; Y A's: 476A 7648 8657 135A 295A; tx's: 1239; Aig: (5,10) P Ref:Ay Exts=l Dual=Aig Elts': 27 13 22 7 9 = 3 ACT # of Cocys: (3,7,10,6,3,2,0,0); # of Cys: (3,7,10,6,3,2,0,0); # of R-psorbs=19; C-psorbs:46 7A 18 2 3 59; # of Bases=141; A's: 14A -37A 678; /)\'s: 568 149 -259; Y A's: 8657 149A; A 2 0 : (5,10) P Ref:Ay Exts=2 Dual=Aiy Elts': 27 13 22 7 11 = 3 DEC # of Cocys: (3,6,11,8,1,1,1,0); # of Cys: (2,10,8,4,6,1,0,0); # of R-psorbs=15; C-psorbs:58 6 1239 4 7A; # of Bases=144; 228 Decomposition info: BTS (5678A,12349), of sizes (5,5) and ranks (3,4), respectively. By \7 /l get minor 8 13 11 5 1 2 4 3 = W4 Summands: Left= G_g. Rights Nj. A's: 56A 678; /^'s: 568 149 234; YA'a: 568A 8657; tx' a: 1239; N 21: (5,10) P Ref:A 7 Exts=l Dual=Y2 i Bits': 27 13 22 7 14 /? = 3 DEC # of Cocys: (1,10,11,4,3,1,1,0); # of Cys: (1,10,11,4,3,1,1,0); # of R-psorbs=ll; C-psorbs: 6 7 12349A 5 8 ; ^ of Bases=160; Decomposition info: FCBTS (149A,235678) Summands: Left= Fj by \2/3/5. Rights IV 5 by / I . A's: 678; /\!s: 568; Y A's: 8657; ex' a: 149A 234A 1239; Y 2 2 : (5,10) P Ref:Y 7 Dual=Y27 Bits': 27 13 22 7 24 /?= 3 DEC # of Cocys: (0,16,0,12,0,3,0,0); # of Cys: (2,7,12,7,2,0,0,1); # of R-psorbs=5; C-psorbs:1239 45678A; # of Bases=156; Decomposition info: FCBTS (1239,45678A) Summands: Left= Fj by \4/5/A. Right= G_n by /I. A's: 45A 678; tx's: 1239; ©'s : 45A678; Y 2 3 : (5,10) P Ref:A 7 Dual=Y23 Bits': 27 13 22 7 25 /?= 4 # of Cocys: (2,9,9,6,4,0,1,0); # of Cys: (2,9,9,6,4,0,1,0); # of R-psorbs=ll; C-psorbs : 6 3459 1 278A; ^ of Bases=149; A's: -26A -678; A's: -149 -135; IV2 4 : (5,10) P Ref:Y 7 Dual=iV2 4 Elts': 27 13 22 7 29 /? = 3 DEC # of Cocys: (2 ,8 ,12,4,2,3,0,0);# of Cys: (2,8,12,4,2,3,0,0); # of R-psorbs—11; C-psorbs:568A 4 1239 7; # of Bases=152; Decomposition info: BTS (5678A,12349), of sizes (5,5) and ranks (3,4), respectively. By \5 /l get minor 13 6 11 14 1 2 4 3 = IV4 Summands: Left= Y5 . Right = Nj. A's: 57A 678; /[!s: 149 234; tx 's : 568A 1239; N 2 5 : (5,10) P Reî:Ng Exts=l Dual=Y25 Bits': 27 29 22 7 6 /? = 3 DEC # of Cocys: (4 ,6, 8 , 8 , 4 , 1,0,0);# of Cys: (4 ,6, 8 , 8 ,4 ,1,0,0); # of R-psorbs=6 ; C-psorbs: 12356789 4 A; # of Bases=136; Decomposition info: BTS (5678A,12349), of sizes (5,5) and ranks (3,4), 2 2 9 respectively. By \A/4 get minor 8 11 13 14 1 2 4 7 = iV4 Summands: Left= Nq. Right = Ng. A's; 19A 23A 58A 67A; /^s: 467 149 234 458; Y A's; 764A 194A 234A 854A; m 's: 1679 5678 2358 1589 2367 1239; N2q: (5,10) P RefriVs Dual=A^26 Bits': 27 29 22 7 14 ,3 - 3 DEC # of Cocys: (0,10,16,0,0,5,0,0); # of Cys: (0,10,16,0,0,5,0,0); ^ of R-psorbs=3;# of C-psorbs=l; ^ of Bases=176; Decomposition info: FCBTS (1679,23458A) Summands: Left= Fj by \2/3/4. Right = N q by / I . m ' s : 1679 467A 5678 149A 2358 1589 234A 2367 458A 1239; A 2 7 : (5,10) D Ref:A^22 Exts=2 Bits': 11 13 14 19 21 ^3 = 3 DEC N28- (5,11) P Ref:Ni7 E x ts = 8 Duai^A^e Bits': 27 13 22 7 3 5 /? = 3 DEC # of Cocys: (2,4,8 , 8 , 6 ,3,0,0,0); # of Cys: (6,10,14,16,10,5,2,0,0); 7^ of R-psorbs=19; C-psorbs:1239A 7 48 5 6 B; of Bases=228; Decomposition info: BTS (1239AB,45678), of sizes (6,5) and ranks (3,4), respectively. By \1\B /5 get minor 24738 11 13 6 = W4 Summands: Left= Ag. Right = G_g. A's: 12A 13B 29B 39A 47B 678 ; 467 568; YA'a: 476B 7648 8657; ex' a: 1239; A 2 9 : (5,11) P Ref:An Exts=l Dual^A^y Bits': 27 13 22 7 3 11 3 = 3 DEC # of Cocys: (1,6 , 7 , 8 , 7,1,1,0,0); # of Cys: (5,11,15,15,11,4,1,1,0); ^ of R-psorbs=19; C-psorbs : 6 123479 5 8 A B; of Bases=240; Decomposition info: BTS (45678B,1239A), of sizes (6,5) and ranks (4,3), respectively. By /5\B \1 get minor 8 11 13 6 2 4 7 3 = W4 Summands: Left= G-j. R ig h ts A 5 . A's: 12A 39A 4AB 56B 678; A'a: 568; YA'a: 568B 8657; m ' a: 1239; Ago: (5,11) P R ef:A i 7 Exts=l Duai=A4 g Bits': 27 13 22 7 3 14 3 = 3 DEC # of Cocys: (1,4,11,8,3,3,1,0,0);# of Cys: (4,12,18,12,8,7,2,0,0); # of R-psorbs=19; C-psorbs : 6 A 7 1239 4 5 8 B; # of Bases—256; Decomposition info: BTS (45678B,1239A), of sizes (6,5) and ranks (4,3), 230 respectively. By \4 /5 \l get minor 11 13 6 14 2 4 7 3 = W4 Summands: Left= A^’ig. Right = N^. A's: 12A 39A 678 -7AB; /f^'s: 568; Y A's: 8657; m 's: 1239; N31 : (5,11) P Ref:Yi7 Dual-Y^g Bits': 27 13 22 7 3 24 ^3 - 3 DEC # of Cocys: (0,8,8,4,8,3,0,0,0);# of Cys: (5,9,17,19,7,2,3,1,0); # of R-psorbs=9; C-psorbs: 6 B 1239 4578 A; # of Bases=252; Decomposition info: BTS (45678B,1239A), of sizes (6,5) and ranks (4,3), respectively. By \4 /5 \l get minor 11 13 682473 = W4 Summands: Left= G_ 4 . Right = A 5 . A's: 12A 39A 45B 678 - 6 AB; tx i's: 1239; ©'s : 45B678; N32 : (5,11) P Ref:#i7 DuaJ=Y5o Bits': 27 13 22 7 3 29 /3 = 3 DEC # of Cocys: (0,6,12,4,4,5,0,0,0);# of Cys: (4,10,20,16,4,5,4,0,0); # of R-psorbs=8 ; C-psorbs:7A 1235689B 4; # of Bases=264; Decomposition info: BTS (45678B,1239A), of sizes (6,5) and ranks (4,3), respectively. By /4\5\1 get minor 11 5 14 13 2 4 7 3 = W4 Summands: Left= Nij. Right = N 3 . A's: 12A 39A 57B 678; m ' s : 568B 1239; N33 : (5,11) P Ref:iVi8 Bxts=l Dual=A 5i Bits': 27 13 22 7 5 6 /) = 3 DEC # of Cocys: (2,6,4,8,10,1,0,0,0); # of Cys: (7,9,11,19,13,2,1,1,0); # of R-psorbs=12; C-psorbs:78 AB 1239 45 6 ; # of Bases=216; Decomposition info: BTS (45678B,1239A), of sizes (6,5) and ranks (4,3), respectively. By /5 \ B \1 get minor 8 11 13 62475 = W4 Summands: Left= G_g. Rights A 5 . A's: 13A 19B 23B 29A 47A 58B 678; /|\'s: 467 568; Y A's: 476A 7648 58GB 8657; txi's: 1239; N34 : (5,11) P R ef:Y i8 Bxts=l Dual=7V52 Bits': 27 13 22 7 5 17 /3 = 3 DEC # of Cocys: (2,4,8,8,6,3,0,0,0); # of Cys: (6,10,14,16,10,5,2,0,0); # of R-psorbs=19; C-psorbs:89B 34 2 17 56 A; # of Bases=226; Decomposition info: BTS (12359B,4678A), of sizes (6,5) and ranks (4,3), respectively. By /2\9\8 get minor 12894 13 73S W4 Summands: Left= N\s. Rights G_g. A's: 13A 15B 29A 47A 678 -89B ; /|\'s: 467 135; Y A's: 476A 7648 135A 153B; 231 7V35: (5,11) P Ref:iVi8 E xts=2 D ual=N s 3 Elts': 27 13 22 7 5 19 ^ = 3 DEC # of Cocys: (3 , 2 , 7 , 1 2 ,5,1,1,0,0); # of Cys: (6,12,12,12,14,7,0,0,0); ^ of R-psorbs=15; C-psorbs : 6 8 B 1239 47 5 A; ^ of Bases=220; Decomposition info: BTS (12359B,4678A), of sizes (6,5) and ranks (4,3), respectively. By \l/5 \8 get minor 24738 11 13 5 = W4 Summands: Left= Nig. Right = G_g. A's: 13A 29A 46B 47A 678 8 AB ; /f^'s: 467 135 259; Y A 's: 467B 476A 7648 135A 295A; m 's: 1239; Nse: (5,11) P Ref:iVi8 Exts=l Dual=Y 54 Elts': 27 13 22 7 5 21 /) = 3 DEC # of Cocys: (1,6,7,8,7,1,1,0,0);# of Cys: (5,11,15,15,11,4,1,1,0); # of R-psorbs=ll; C-psorbs:123589B 4 6 7 A; # of Bases=244; Decomposition info: BTS (45678B,1239A), of sizes (6,5) and ranks (4,3), respectively. By \5/ B \1 get minor 4 7 12 11 1298 = W4 Summands: Left= Nig. Right = Y 5 . A's: 13A 29A 47A 5AB 678; Vs: 467; Y A's: 476A 7648; m 's: 135B 259B 1239; Nsj: (5,11) P Ref:Yi8 Dual=Y 55 Elts': 27 13 22 7 5 29 /? = 3 DEC # of Cocys: (0 , 10, 0,16, 0,5,0,0,0); # of Cys: (5,10,16,16,10, 5, 0,0,1); # of R-psorbs=6 ; C-psorbs:12345689B 7A; # of Bases=248; Decomposition info: BTS (45678B,1239A), of sizes (6,5) and ranks (4,3), respectively. By \5/B \1 get m inor 4 3 8 11 1 2 13 12 = W4 Summands: Left= Nig. Right = Y 5 . A's: 13A 29A 47A 57B 678; m ' s : 568B 1239; #3 8 : (5,11) P Ref:#i8 Dual= # 5 6 Elts': 27 13 22 7 5 30 /? = 3 DEC # of Cocys: (2,4 ,8 , 8 , 6,3,0,0,0); # of Cys: (6,10,14,16,10,5,2,0,0); # of R-psorbs=12; C-psorbs:4678B 1239 A 5; # of Bases=232; Decomposition info; BTS (12359A,4678B), of sizes (6,5) and ranks (4,3), respectively. By /1 \ A \4 get m inor 1 2 8 3 13 6 11 15 = W4 Summands: Left= Nig. R ight= #5. A's: 13A 29A 47A 48B 678 6 AB ; /^'s: 135 259; Y A's: 135A 295A; m 's: 467B 1239; #39 : (5,11) P Ref:#i9 Exts=l Dual= # 5 7 Elts': 27 13 22 7 9 11 /? = 3 DEC # of Cocys: (2,3,9,10,4,2,1,0,0); # of Cys: (5,12,15,12,11,7,1,0,0); 232 # of R-psorbs=23; C-psorbs:7B 23 9 1 6 A 458; ^ of Bases=235; Decomposition info: BTS (12349A,5678B), of sizes (6,5) and ranks (4,3), respectively. By /1\ A \7 get minor 12438 13 11 5 = W4 Summands: Left= Nig. Right = G-g. A's: 14A -2AB -37A 56B 678; 568 149; Y A's: 568B 8657 149A; N40: (5,11) P Ref:Y20 Exts=l Dual^jVgg Bits': 27 13 22 7 11 14 = 3 DEC # of Cocys: (1,3,11,11,3,0,1,1,0); # of Cys: (2,18,14,8,14,5,2,0,0); # of R-psorbs=9; C-psorbs : 6 12349B 58 7A; # of Bases=268; Decomposition info: BTS (12349B,5678A), of sizes (6,5) and ranks (4,3), respectively. By \l/2 \7 get minor 24368 13 10 5 = W4 Summands: Lefts N 21. R ig h ts G-g. A's: 56A 678; 4 \'s: 568; Y A's: 568A 8657; ex' s: 149B 234B 1239; Y 41: (5,11) P Ref:Y20 Bnal=Nsg Bits': 27 13 22 7 11 29 /? = 3 DEC # of Cocys: (2,2 ,1 0 ,1 2 , 2 , 1 , 2 ,0,0); # of Cys: (4,14,16,8,12,9,0,0,0); # of R-psorbs=10; C-psorbs:568B 1239 4 7A; # of Bases=248; Decomposition info: BTS (5678AB,12349), of sizes (6,5) and ranks (3,4), respectively. By \5\7/l get minor 13 11 5 14 1 2 4 3 S W4 Summands: Lefts Ng. R ig h ts Nj. A's: 56A 57B 678 8 AB; 4 \'s: 149 234; ex' s: 568B 1239; #4 2 : (5,11) P Ref:#2l DuaJ= # 6 0 Bits': 27 13 22 7 14 29 /? = 3 DEC # o f Cocys: (0,4,14,8,0,3,2,0,0); # o f Cys: (2,14,22,8,6,9,2,0,0); # of R-psorbs=7; C-psorbs:568B 7 12349A; # of Bases=288; Decomposition info: BTS (12349A,5678B), of sizes (6,5) and ranks (4,3), respectively. By \l/2 \5 get minor 2 4 3 6 13 7 10 15 S W4 Sum m ands: Lefts #2 1 - Rights # 5 . A's: 57B 678; ex' s: 568B 149A 234A 1239; #43: (5,11) P Ref:#2,5 D u a l= #6i Bits': 27 29 22 7 6 14 ^ = 3 DBC # of Cocys: (0,10,0,16,0,5,0,0,0); # of Cys: (5,10,16,16,10,5,0,0,1); # of R-psorbs=3; C-psorbs:123456789B A; # of Bases=256; Decomposition info: BTS (14589B,2367A), of sizes (6,5) and ranks (4,3), respectively. By \l/9 \2 get minor 4 8 11 7 2 14 13 3 S # 3 Summands: Lefts #2 5 . R ig h ts Nq. 233 A's: 19A 23A 4AB 58A 67A; txi' 5 : 1679 467B 5678 149B 2358 1589 234B 2367 458B 1239; N 44: (5,11) P Ref:#27 Exts= 2 Dual=#62 Elts': 11 13 14 19 21 3 /? = 3 DEC # of Cocys: (2,3,8 ,1 1 , 6 ,0 ,0 , 1 ,0); # of Cys: (4,16,10,12,16,3,2 , 0 , 0 ); # of R-psorbs=ll; C-psorbs:B 1259 3A 4678; # of Bases=240; Decomposition info: BTS (12359A,4678B), of sizes (6,5) and ranks (4,3), respectively. By \l/3 \4 get minor 2 8 11 97563 = W4 Summands: Left= G_ 5 . Right = # 5 . A's: 12B 46B 59B 78B; J^'s: 59A 123; YA 'a: 59AB 123B; tx' a: 4678; ®'s : 59A123; #4 5 : (5,11) P Ref:#27 Exts=l Dual= # 6 3 E lts': 11 13 14 19 21 7 /I = 3 DEC # of Cocys: (1,2,12,12,2,1,0,0,1);# of Cys: (0,26,0,24,0,13,0,0,0); # of R-psorbs=7; C-psorbs:1234678B 59A; # of Bases=280; Decomposition info: FCBTS (4678,12359AB) Summands: Left= Fj by \l/2\3/9. Right= # 9 by /4. /^'s: 59A; x ' a: 4678 123B; #46: (6 11) D Ref: # 2 8 Exts= 17 Elts' : 59 29 46 3 5 /? == 3 DEC #47: (6 11) D Ref; #29 Exts= 12 Elts' ; 59 61 14 35 5 /3 = 3 DEC #48: (6 11) D R ef:#3 o Exts= =6 Elts': 27 61 46 35 5 ^ - = 3 DEC #49 : (6 11) D Ref: # 3 1 Exts= :6 Elts': 27 29 14 35 37 /3 = 3 DEC #50: (6 11) D Ref:#32 Exts= -2 Elts': 59 29 46 35 37 /3 = 3 DEC #51: (6 11) D Ref: # 3 3 Exts= :6 Elts': 27 45 62 3 5 /3 = 3 DEC #52: (6 11) D Ref: # 3 4 Exts= -7 Elts': 59 13 30 3 37 /3 =-- 3 DEC #5 3 : ( 6 11) D Ref: # 3 5 Exts-=9 Elts': 59 45 30 3 37 /3 == 3 DEC #5 4 : ( 6 11) D Ref: #36 Exts= : 6 Elts': 59 13 62 3 37 ^ == 3 DEC #5 5 : ( 6 11) D Ref: # 3 7 Exts= :4 Elts': 59 13 62 35 37 /3 = 3 DEC #56 : ( 6 11) D Ref: #38 Exts= :4 Elts': 27 45 62 35 37 = 3 DEC #5 7 : ( 6 11) D Ref; # 3 9 Exts= =1 E lts'; 59 45 14 5 1 5 /3 == 3 DEC #58: ( 6 11) D R ef:#4 o Exts= =1 Elts': 27 61 46 51 5 /3 == 3 DEC #5 9 : ( 6 11) D Ref: #41 Elts': 59 29 46 51 37 /3 = 3 DEC #60 : ( 6 11) D Ref:#42 Exts= -1 Elts': 43 29 62 51 37 (3 = 3 DEC #61 : ( 6 11) D Ref: # 4 3 Exts= --2 Elts': 11 61 62 35 7 /3 == 3 DEC 234 Nq2- (6,11) D Ref:iV44 Elts': 59 45 22 7 24 /? = 3 DEC #63: (6,11) D Ref:#45 Elts': 59 45 54 7 24 /? = 3 DEC #64: (5,12) PRef :#28 D u a l= # g4 Elts': 27 13 22 7 3 5 6 ^ = 3 DEC # of Cocys: (2 ,2 ,4 , 8 ,10, 5,0,0,0,0); # of Cys: (10,16,18,30,30,15,6,2, 0,0); # of R-psorbs—1 2 ; C-psorbs:1239 45 6 78 A BC; ^ of Bases=344; Decomposition info: BTS (45678C,1239AB), of sizes ( 6 , 6 ) and ranks (4,3), respectively. By /5\ C \1 \ A get minor 8 11 13 62475 = M'^4 Summands: Left= G_ 6 - Right = # 9 . A's: 12A 13B 19C 23C 29B 39A 47B 58C 678 ABC; /^'s: 467 568; Y A's: 476B 7648 586C 8657; m 's: 1239; #65: (5,12) P Ref:#28 Dual= # 8 5 Bits': 27 13 22 7 3 5 11 ^3 = 3 DEC # of Cocys: (1,2,7,8 ,7,5,1,0,0,0); # of Cys: (8,16,24,30,24,15,8,2,0,0); ^ of R-psorbs=15; C-psorbs : 6 AB 123479C 58; ^ of Bases=380; Decomposition info: BTS (45678C,1239AB), of sizes ( 6 , 6 ) and ranks (4,3), respectively. By /5 \C \1 \A get minor 8 11 13 6 2 4 7 5 = M' 4 Summands: Left= G_y. Right = # 9 . A's: 12A 13B 29B 39A 47B 4AC 56C 678; A'a: 568; Y A's: 568C 8657; m 's: 1239; #6 6 : (5,12) P Ref:#28 D u a l= # g6 Elts': 27 13 22 7 3 5 14 /) = 3 DEC # of Cocys: (1,2,5,12,7,1,3,0,0,0); # of Cys: (7,18,26,24,24,21,6,0,1,0); of R-psorbs=15; C-psorbs:1239 4C 5 6 7 8 AB; ^ of Bases=396; Decomposition info: BTS (45678C,1239AB), of sizes ( 6 , 6 ) and ranks (4,3), respectively. By \4 /5 \l\ A get minor 11 13 6 14 2 4 7 5 = W4 Summands: Left= #1 9 . Right = # 9 . A's: 12A 13B 29B 39A -47B 678 -7AC; ^'s: 568; Y A's: 8657; txi's: 1239; #6 7 : (5,12) P Ref: # 2 8 D u a l= # g7 Elts': 27 13 22 7 3 5 19 ^ = 3 DEC # of Cocys: (1,4,3,8,11,3,1,0,0,0); # of Cys: (9,16,20,30,30,15,4,2,1,0); # of R-psorbs=19; C-psorbs : 6 B 8 C 1239 4 5 7 A; of Bases=360; Decomposition info: BTS (45678C,1239AB), of sizes ( 6 , 6 ) and ranks (4,3), respectively. By /5\ C \1\B get minor 8 11 13 62473 = W4 Summands: Left= G_ 7 - R ight= # 9 . 235 A'a: 12A 13B 29B 39A 46C 47B 5AC 678 SBC; /f!s: 467; Y A'a: 467C 476B 7648; M' a: 1239; Yes: (5,12) P Ref:iV28 Dual=jVgg Elts': 27 13 22 7 3 5 21 ^ = 3 DEC # of Cocys: ( 1 ,2 ,7 ,8 , 7,5,1,0,0,0); # of Cys: (8 , 1 6 ,2 4 ,3 0 ,2 4 ,15,8,2,0,0); ^ of R-psorbs=19; C-psorbs : 6 A B 12389C 4 7 5 ;^ of Bases=384; Decomposition info: BTS (45678C,1239AB), of sizes ( 6 , 6 ) and ranks (4,3), respectively. By /5\ C \1\ A get minor 8 11 13 62475 = W4 Summands: Left= Yjg. Right = 77g. A's: 12A 13B 29B 39A 47B SBC 678 - 8 AC; a'-s: 467; Y A's: 476B 7648; txi' a: 1239; Nqq: (5,12) P Ref:#28 Dual=Ygg Elts': 27 13 22 7 3 5 24 /3 = 3 DEC # of Cocys: (0,4,8,4,8,7,0,0,0,0); # of Cys: (8,14,24,36,24,9,8,4,0,0); # of R-psorbs=9; C-psorbs:AB 1239 467C 58; # of Bases=396; Decomposition info: BTS (45678C,1239AB), of sizes ( 6 , 6 ) and ranks (4,3), respectively. By \4 /5 \l\ A get minor 11 13 682475 = W4 Summands: Left= G- 4 . Right = Ng. A's: 1 2 A 13B 29B 39A 45C -47B 678 - 6 AC; txi' a: 1239; ©'a : 45C678; Njq: (5,12) P Ref:#28 Dual=Ægo Elts': 27 13 22 7 3 5 29 /? = 3 DEC # of Cocys: (0,4,6,8,8,3,2,0,0,0); # of Cys: (7,16,26,30,24,15,6,2,1,0); # of R-psorbs=1 2 ; C-psorbs:1239 B 7 4568C A; ^ of Bases=408; Decomposition info: BTS (45678C,1239AB), of sizes ( 6 , 6 ) and ranks (4,3), respectively. By \5/ C \1 \ A get minor 4 3 8 11 1 2 13 12 = W4 Summands: Left= A^ig. Right = Ng. A's: 12A 13B 29B 39A 47B 57C 678; txi' a: 568C 1239; N71: (5,12) P Ref:N2g Dual=Ngi Elts': 27 13 22 7 3 5 30 ^ = 3 DEC # of Cocys: (0,6,0,16,0,9,0,0,0,0); # of Cys: (8,15,2 4 ,32,24,15,8,0,0,1); ^ of R-psorbs= 8 ; C-psorbs:12346789AC 5 B; of Bases=392; Decomposition info: BTS (4678BC,12359A), of sizes ( 6 , 6 ) and ranks (3,4), respectively. By \4\ B \l/5 get minor 11 13 6 14 2473 = W4 Summands: Left= Ng. Right = Nij. A's: 12A 13B 29B 39A 47B 48C 678 6 BC; txi' a: 467C 1239; 236 N^2■ (5,12) P Ref:#29 DuaI=A^ 9 2 Elts': 27 13 22 7 3 11 14 /3 - 3 DEC # of Cocys: (1 ,1,7,11,7,2,1,1,0,0); # of Cys: (6,20,26,22,2 6 ,19,6,2,0,0); # of R-psorbs—15; C-psorbs:A 6 1239 4C 58 7B; # of Bases=408; Decomposition info: BTS (45678BC,1239A), of sizes (7,5) and ranks (4,3), respectively. By \4/5\ B \1 get minor 11 13 6 14 2 4 7 3 = W4 Summands: Lefts Rights N^. A's: 12A 39A -4AB 56B 678 -7AC ; /^'s: 568; Y A's: 568B 8657; tx's: 1239; Y 7 3 : (5,12) P Ref:#3 o Dual^A'gg Elts': 27 13 22 7 3 14 29 /? = 3 DEC # of Cocys: (0,2,8 , 1 2 ,4,1,4,0,0,0); # of Cys: (5,18,32,2 4 ,18,21,8,0,1,0); # of R-psorbs=8 ; C-psorbs:1235689C 7A 4 B; # of Bases=448; Decomposition info: BTS (45678BC,1239A), of sizes (7,5) and ranks (4,3), respectively. By \4\5/ C \1 get minor 3 8 11 7 1 2 13 15 S Summands: Lefts N-^q- R ig h ts N^. A's: 12A 39A 57C 678 -7AB; xi's: 568C 1239; W74: (5,12) P Ref:iV3 3 Dual^Yg^ Elts': 27 13 22 7 5 6 29 ^3 = 3 DEC # of Cocys: (0,6 ,4 ,4 ,12,5 ,0 ,0 ,0,0); # of Cys: (9 , 14,20,36, 30,9, 4,4,1,0); # of R-psorbs=8 ; C-psorbs:7A 1235689C 4 B; 7^ of Bases=376; Decomposition info: BTS (45678BC,1239A), of sizes (7,5) and ranks (4,3), respectively. By \5\7/8\l get minor 478 15 10 2 9 3 S W4 Summands: Lefts A^gg. Rights N^. A's: 13A 19B 23B 29A 47A 57C 58B 678 6 BC; xi' a: 568C 1239; A^7 5 : (5,12) P Ref: # 3 4 Dual^A^gs Elts': 27 13 22 7 5 17 19 /I = 3 DEC # of Cocys: (2,1,5,10,8,4,1,0,0,0); # of Cys: (9,17,21,27,27,18,7,1,0,0); # of R-psorbs=23; C-psorbs:B A 29 3 147 5 6 8 C; ^ of Bases=353; Decomposition info: BTS (12359BC,4678A), of sizes (7,5) and ranks (4,3), respectively. By \1/5\B \ 8 get minor 2 4 7 3 8 11 13 5 S W4 Summands: Lefts A/g^. Rights G_g. A's: 13A 15B 29A -2BC 46C 47A 678 -89B 8 AC; /^'s: 467 135; Y A's: 467C 476A 7648 135A 153B; Njq: (5,12) P Ref: # 3 5 Dual^Ygf, Elts': 27 13 22 7 5 19 21 /? = 3 DEC # of Cocys: (1,3,3,11,11,0,1,1,0,0); # of Cys: (7,20,22,22,32,19,2,2,1,0); # of R-psorbs=9; C-psorbs : 6 8 B 12359C 47 A; 7^ of Bases=388; 237 Decomposition info: BTS (12359BC,4678A), of sizes (7,5) and ranks (4,3), respectively. By \l\2 /3 \8 get minor 8 3 11 9 4 15 5 1 = W4 Summands: Left= Ægg. Right = G-g. A's: 13A 29A 46B 47A 5AC 678 8AB; 467; Y A's: 467B 476A 7648; M's: 135C 259C 1239; Njt. (5,12) P Ref:iV3 5 Dual^Æg? Bits': 27 13 22 7 5 19 30 /? = 3 DEC # of Cocys: (2,2,2,12,10,1,2,0,0,0); # of Cys: (9,18,20,24,30,21,4,0,1,0); ^ of R-psorbs=10; C-psorbs:1239 467C 8B A 5; # of Bases=:360; Decomposition info: BTS (467ABC,123589), of sizes (6,6) and ranks (3,4), respectively. By \4 \ A /1\3 get minor 13 6 9 15 1 8 11 3 S W4 Summands: Left= Nc). Right = N'2q. A's: 13A 29A 46B 47A 48C 678 6AC 7BC 8AB; ^ s: 135 259; Y A's: 135A 295A; tx's: 467C 1239; #7 8 : (5,12) P Ref: # 3 6 Dua]=#g8 Bits': 27 13 22 7 5 21 30 /I = 3 DBC # of Cocys: (0,4,6,8,8,3,2,0,0,0); # of Cys: (7,16,26,3 0 ,2 4 ,1 5 ,6 ,2 ,1 ,0 ); of R-psorbs=7; C-psorbs:467C 12359B 8 A; of Bases=416; Decomposition info: BTS (45678BC,1239A), of sizes (7,5) and ranks (4,3), respectively. By \4\5/ B \1 get m inor 7 12 11 15 1 2 9 8 = IV4 Summands: Left= #3 8 - Right = # 5 . A's: 13A 29A 47A 48C 5AB 678 6 AC; s: 467C 135B 259B 1239; #7 9 : (5,12) P Ref: # 3 9 D ual=#gg Bits': 27 13 22 7 9 11 15 = 3 DBC # of Cocys: (2 ,0 ,6, 12,6, 3 , 2 ,0,0,0); # of Cys: (8 ,1 8 , 2 4 ,2 4 ,2 4 ,21,8,0,0,0); # of R-psorbs=12; C-psorbs:46 7ABC 1589 23; # of Bases=368; Decomposition info: BTS (12349AC,5678B), of sizes (7,5) and ranks (4,3), respectively. By /1\2\ A \7 get minor 24378 13 11 5 = IV4 Summands: Left= #39. Right = G _ 9 . A's: 14A -27C -2AB -37A -3BC 49C 56B 678; /|v's: 568 149; Y A's: 568B 8657 149A 941C; #80: (5,12) P Ref:#4 o Dual=#ioo Bits':27 13 22 7 11 14 29 /I = 3 DEC # of Cocys: (0,4,0,22,0,3,0,2 ,0,0); # of Cys: (4,23,2 8 , 16,2 8 ,2 3 , 4 ,0 ,0 ,1 ); ^ of R-psorbs= 6 ; C-psorbs:12349B 568C 7A; ^ of Bases=448; Decomposition info: BTS (5678AC,12349B), of sizes ( 6 , 6 ) and ranks (3,4), 238 respectively. By \5 \7 \l/2 get minor 13 10 5 15 2 4 3 6 = VF4 Summands: Left= Ng. R ig h ts # 2 1 - A's: 56A 57C 678 SAC; m ' s : 568C 149B 234B 1239; #81 : (5,12) P Ref: # 4 4 Dual=#ioi Elts': 11 13 14 19 21 3 5 /3 = 3 DEC # of Cocys: (2,1,4,11,10,2,0,1,0,0); # of Cys: (8,20,20,22,32,19,4,2,0,0); # of R-psorbs=:ll; C-psorbs:BC 239A 15 4678; 7^ of Bases=360; Decomposition info: BTS (4678BC,12359A), of sizes ( 6 , 6 ) and ranks (3,4), respectively. By \4\B \l/2 get minor 5763289 11 = W4 Summands: Left= Ng. Right = G_ 5 . A's: 12B 13C 46B 47C 59B 5AC 6 8 C 78B; /(\'s: 59A 123; Y A's: 59AB 5A9C 123B 132C; txi's: 4678; ©'s : 59A123; #82 : (5,12) P Ref:# 4 4 D u a l= # io2 Elts': 11 13 14 19 21 3 7 /? = 3 DEC # of Cocys: (1,2,4,1 2 ,10,1, 0,0,1,0); # of Cys: (5, 26,16,24,34,13,8 , 0,1,0); # of R-psorbs=9; C-psorbs:B 1234678C 59 A; ^ of Bases=408; Decomposition info: BTS (12359AC,4678B), of sizes (7,5) and ranks (4,3), respectively. By \l/2 \3 \4 get minor 8 9 11 35761 = W4 Summands: Left= #4 4 . Right = # 5 . A 's: 12B 3BC 46B 59B 78B; /|\'s: 59A; Y A's: 59AB; x 's : 4678 123C; #83: (5,12) P Ref:# 4 5 D u a l= # io3 Elts': 11 13 14 19 21 7 22 /? = 3 DEC # of Cocys: (0,3,0,24, 0, 3,0,0,0 , 1 ); # of Cys: (0, 39,0,48,0,39,0,0,0,1); # of R-psorbs=4;# of C-psorbs=l; # of Bases=480; Decomposition info: FCBTS (4678,12359ABC) Summands: Left= Fj by \l/2\3\5/A . Right = # 1 5 by /4 . txi's: 4678 59AC 123B; # 8 4 (7,12) D R ef: # 6 4 Elts': 59 93 110 3 5 /I = : DEC # 8 5 (7,12) D R ef: # 6 5 Elts': 123 93 46 67 5 /? = 3 DEC # 8 6 (7,12) D R ef: # 6 6 Elts': 59 93 110 67 5 /? = 3 DEC # 8 7 (7,12) D R ef: # 6 7 Elts': 123 93 46 3 69 ^ = 3 DEC #88 (7,12) D R ef: # 6 8 Elts': 123 29 110 3 69)8 = = 3 DEC # 8 9 (7,12) D R ef: # 6 9 Elts': 59 29 46 67 69 /? = 3 DEC # 9 0 (7,12) D R ef: #7 0 Elts': 123 29 110 67 69 (5= 3 DEC 239 A^9 1 : (7,12) D Ref:AT? 1 Bits': 59 93 110 67 69 = 3 DEC #9 2 : (7,12) D Ref:# 7 2 Bits': 59 125 78 99 5 /9 = 3 DEC #9 3 : (7,12) D Ref: # 7 3 Bits': 91 61 110 99 69 ^ = 3 DEC #9 4 : (7,12) D Ref: # 7 4 Bits': 91 45 126 67 69 ^ = 3 DEC #9 5 : (7,12) D Ref: # 7 5 Bits': 123 77 30 3 101 /3 = 3 DEC #9 6 : (7,12) D Ref:#y6 Bits': 123 45 94 3 101 /3 = 3 DEC #9 7 : (7,12) D Ref: # 7 7 Bits': 59 109 94 67 101 ^ = 3 DEC #9 8 : (7,12) D Ref: # 7 8 Bits': 59 77 126 67 101 /? = 3 DEC # 9 9 : (7,12) D Ref:#7 g Bits': 123 109 78 115 5 /? = 3 DEC # 1 0 0 : (7,12) D Ref: # 8 0 Bits': 91 61 110 115 69 /) = 3 DEC #101: (7,12) D Ref: # 8 1 Bits': 123 45 86 7 24 /? = 3 DEC #102: (7,12) D Ref:#s2 Bits': 123 109 86 7 24 /) = 3 DEC # 1 0 3 : (7,12) D Ref: # 8 3 Bits': 59 109 118 7 88 /) = 3 DEC # 1 0 4 : (6,12) P Ref:#4 6 D u a l= #6 i 4 Bits': 59 29 46 3 5 6 /) = 3 DEC # of Cocys: (5,5,9,19,15,6,3,1,0,0); # of Cys: (4,8,10,12,16,11,2,0,0,0); # of R-psorbs=25; C-psorbs:A 12 3 4789 56 BC; # of Bases=380; Decomposition info: BTS (46789C,1235AB), of sizes ( 6 , 6 ) and ranks (4,4), respectively. By /4\ C /5\B get minor 8 11 5 14 1243 = W4 Summands: Left= #2 0 . Right = G_ 7 . A's: 12A 13B 23C ABC; J[!s: 578 679 459 468 123; Y A's: 123A 132B 321C; cx's: 4789; # 1 0 5 : (6,12) P Ref: # 4 6 D u a l= #0 i 5 Bits': 59 29 46 3 5 7 ^ = 3 DBC # of Cocys: (4,6,12,16,12,9,4,0,0,0); # of Cys: (4,6,12,16,12,9,4,0,0,0); ^ of R-psorbs=24; C-psorbs:45789 123AC 6 B; of Bases=408; Decomposition info: BTS (45789B,1236AC), of sizes ( 6 , 6 ) and ranks (4,4), respectively. By /4\ B \ l / 6 get minor 8 11 13 62437 = W4 Summands: Left= #2 0 - R ight= # 1 7 . A's: 12A 13B 2BC 3AC; 578 679 459 468; m ' s : 4789 123C; #106: (6,12) P Ref: # 4 6 D u a l= # io6 Bits': 59 29 46 3 5 13 /? = 3 DEC # of Cocys: (4,6,12,16,12,9,4,0,0,0); # of Cys: (4,6,12,16,12,9,4,0,0,0); # of R-psorbs=63; C-psorbs:358A 7Cl2469 B;T(4of Bases=394; 240 Decomposition info: BTS (45789C,1236AB), of sizes ( 6 , 6 ) and ranks (4,4), respectively. By \4/7/2\ A get minor 4 9 11 13 14 1 8 15 = W4 Summands: Left= Niq. Right = G-j. A's: 12A 13B -4BC 58C; /|\'a: 578 -679 26A 123; Y A 's: 587C 2A61 123A 132B; A^IO?: (6,12) P Ref: A/)g D u a l= A j 2 g Elts': 59 29 46 3 5 14 /3 = 3 DEC # of Cocys: (3, 7,13,15,13,8,3,1, 0,0); # of Cys: (3, 7,13,15,13,8 ,3 ,1 ,0 , 0); # of R-psorbs—55; C-psorbs:l 29 3 4 58 6 7 AC B; # of Bases—418; Decomposition info: BTS (46789C,1235AB), of sizes ( 6 , 6 ) and ranks (4,4), respectively. By \4/7/2\ A get minor 8 9 11 3 14 1 4 15 S W4 Summands: Left= A 1 9 . Right = G-j. A's: 12A 13B 69C; /t!s: -578 679 123; Y A's: 697C 123A 132B; Aios: (6,12) P Ref:A46 D u a l= A i3 2 E lts': 59 29 46 3 5 19 /3 = 3 DEC # of Cocys: (3,7,13,15,13,8,3,1,0,0); # of Cys: (3,7,13,15,13,8,3,1,0,0); ^ of R-psorbs=35; C-psorbs:3 A 1 2 4789 6 5BC; # of Bases—420; Decomposition info: BTS (46789C,1235AB), of sizes ( 6 , 6 ) and ranks (4,4), respectively. By /4\8/2\ A get minor 8 13 10 51243 = W4 Summands: Left= N 2Q. Right = A 's: 12A 13B 5AC; A's: 679 468 123; Y A 's: 123A 132B; 1x 1' s: 4789; A 109 : (6,12) P Ref:A4G D u al= A]22 Elts': 59 29 46 3 5 21 /3 = 3 DEC # of Cocys: (4,8,10,12,16,11,2,0,0,0); # of Cys: (5,5,9,19,15,6,3,1,0,0); ^ of R-psorbs=39; C-psorbs:B 4789 12356AC;# of Bases=376; Decomposition info: BTS (46789C,1235AB), of sizes ( 6 , 6 ) and ranks (4,4), respectively. By /4 \ C /2\ A get minor 8 13 7 10 1243 = W4 Summands: Left= N\g. Right = G-j. A's: 12A 13B 48C 5BC 79C; A'a: 679 26A 468 123; Y A's: 976C 2A61 486C 123A 132B; m 's: 4789; Alio: (6,12) P Ref:A 4 G Dual=A iG8 E lts': 59 29 46 3 5 22 /? = 3 DEC # of Cocys: (3,5,15,19,9,6,5,1,0,0); # of Cys: (2,8,16,12,10,11,4,0,0,0); # of R-psorbs=27; C-psorbs : 6 1 23 4789 5C AB; # of Bases=448; Decomposition info: BTS (1235ABC,46789), of sizes (7,5) and ranks (4,4), respectively. By \l/2 \ A /4 get minor 24368 13 7 10 = W4 241 Summands: Left= TVgg. Right = Nj. ùJs\ 12A 13B; A's; 679 468 123; Y A's; 123A 132B; ce's; 4789; A^lli; (6,12) P Ref;iV4 6 Dual=7Vi23 Elts'; 59 29 46 3 5 33 /3 = 3 DEC # of Cocys; (2,10,10,16,14,5,6,0,0,0); # of Cys; (3,6,14,16,12,9,2,0,1,0); # of R-psorbs=22; C-psorbs;l 236AC 4789 5 B; # of Bases—440; Decomposition info; BTS (45789B,1236AC), of sizes (6,6) and ranks (4,4), respectively. By /4 \9 \l/2 get minor 4 13 732819 = BQ Summands; Left= Noq- R ight= N\^. A's: 12A 13B 16C; A'a: 578 459; xi's: 4789 26AC; # 1 1 2 : (6,12) P Ref; # 4 6 D ual=jV i5 2 Elts': 59 29 46 3 5 35 /3 = 3 DEC # of Cocys; (4,6,10,18,16,5,2,2,0,0); # of Cys; (3,9,11,11,17,10,1,1,0,0); # of R-psorbs=25; C-psorbs;lA 2 36 4789 5 BC; # of Bases=396; Decomposition info; BTS (45789C,1236AB), of sizes (6,6) and ranks (4,4), respectively. By /4\9/2\ A get minor 4 13 791283 = W 4 Summands; Left= #20- Right = G-g. A's; 12A 13B 6AC; A'a: 578 26A 459 123; Y A's: 2A61 A62C 123A 132B; tx's; 4789; # 1 1 3 ; (6,12) P Ref; # 4 6 D u a l= #4 i 3 Elts': 59 29 46 3 5 36 /3 = 3 DEC # of Cocys: (2,10,12,10,18,9, 0,2, 0,0); # of Cys: (3, 7,11,17,17,4,1,3,0,0); ^ of R-psorbs=25; C-psorbs;13 2C 4789 5 6 A B; ^ of Bases=432; Decomposition info; BTS (45789B,1236AC), of sizes ( 6 , 6 ) and ranks (4,4), respectively. By /4\9\l/2 get minor 4 13 73281 10 = TT4 Summands; Left= #ig. Right = G_ 4 . A's; 12A -13B 36C; A'a: 578 459; m ' s; 4789; (s)'s ; 12A36C; #1 1 4 ; (6,12) P Ref; # 4 6 Dual=#]% Elts': 59 29 46 3 5 37 ^ = 3 DEC # of Cocys; (3,7,13,15,13,8,3,1,0,0); # of Cys; (3,7,13,15,13,8,3,1,0,0); ^ of R-psorbs=39; C-psorbs;l B 3 4789 5 AC 6 2;:^ of Bases=424; Decomposition info; BTS (45789B,1236AC), of sizes ( 6 , 6 ) and ranks (4,4), respectively. By /4\9/6\ C get minor 8 11 13 51243 = IP 4 Summands; Left= #2 0 - Right = #i§. A's; 12A 13B 6 BC; A'a: 578 459 123; Y A's; 123A 132B; s; 4789; 242 #115: (6,12) P Ref:#4G D u a l= # ii5 Elts': 59 29 46 3 5 38 = 3 DEC # of Cocys: (4,6,12,16,12,9,4,0,0,0); # of Cys: (4,6,12,16,12,9,4,0,0,0); of R-psorbs=39; C-psorbs:34789A 5 C 1 2 6 B; # of Bases=388; Decomposition info: BTS (46789C,1235AB), of sizes ( 6 , 6 ) and ranks (4,4), respectively. By /4\ C /2\ A get minor 8 13 7 10 1243 = lEi Summands: Left= N\g. Right = G-y. A's: 12A 13B 49C 78C; /j\'s: 578 26A 459 123; Y A's: 875C 2A61 495C 123A 132B; m ' s : 4789; #116: (6,12) P Ref:#46 D u a l= #37 i Elts': 59 29 46 3 5 39 /? = 3 DEC # of Cocys: (3,5,15,19,9,6,5,1,0,0); # of Cys: (2,8,16,12,10,11,4,0,0,0); # of R-psorbs=39; C-psorbs:l 6 2 4789 C B 3 5 A; of Bases=448; Decomposition info: BTS (45789B,1236AC), of sizes ( 6 , 6 ) and ranks (4,4), respectively. By /4\9\l/3 get minor 4 15 5 1 2 8 3 11 = W4 Summands: Left= #2 0 - Right = # 1 9 . A 's: 12A -13B; 4 \'s: 578 26A 459; Y A's: 2A61; xi's: 4789; # 1 1 7 : (6,12) P Ref:#46 D u a l= #8 ig Bits': 59 29 46 3 5 48 /? = 3 DEC # of Cocys: (1,13,9,11,19,6,3,1,0,0); # of Cys: (3,6,12,18,16,5,0,2,1,0); # of R-psorbs=25; C-psorbs:AB 23 1 4789 5C 6 ; # of Bases=440; Decomposition info: BTS (456789C,123AB), of sizes (7,5) and ranks (5,3), respectively. By \4/7/8\ A get minor 4 8 2 12 7916 = W4 Summands: Left= Nq2- Right = G_g. A 's: 12A 13B 56C; 4 \'s: 123; Y A's: 123A 132B; txi's: 4789; # 1 1 8 : (6,12) P Ref: # 4 6 D u a l= #2 iS Elts': 59 29 46 3 5 51 /3 = 3 DEC # of Cocys: (2,12,8,12,18,7,4,0,0,0); # of Cys: (4,5,11,19,15,6,1,1,1,0); # of R-psorbs=39; C-psorbs:A 1 B 2 3 4789 5 6 C] ^ of Bases=408; Decomposition info: BTS (46789C,1235AB), of sizes ( 6 , 6 ) and ranks (4,4), respectively. By \4/6/2\ A get minor 13 15 691283 = W4 Summands: Left= #1 7 . Right = G_y. A's: 12A 13B 47C 89C; /K's: 26A 123; Y A's: 2A61 123A 132B; tx's: 4789; # 1 1 9 : (6,12) P Ref: # 4 6 D u a l= #25 i Bits': 59 29 46 3 5 53 ^ = 3 DEC # of Cocys: (1,11,11,15,15,4,5,1,0,0); # of Cys: (2,7,15,15,13,8,1,1,1,0); # of R-psorbs=19; C-psorbs:l B A 2 3 456789C; # of Bases=468; 243 Decomposition info: BTS (46789C,1235AB), of sizes ( 6 , 6 ) and ranks (4,4), respectively. By \4/6/2\ A get minor 13 15 6 11 1 2 8 3 = W4 Summands: Left= A^2 1 - Right = G_y. A's: 12A 13B; /^'s: 123; Y A's: 123A 132B; txi's: 4789 679C 468C; #120: (6,12) P Ref:#4G D n a l^ #2 io Bits': 59 29 46 3 5 54 /? = 3 DEC # of Cocys: (2,8,14,14,14,7,2,2,0,0);# of Cys: (2,8,14,14,14,7,2,2,0,0); # of R-psorbs=23; C-psorbs:l 3A 2 45789C 6 B; # of Bases=456; Decomposition info: BTS (45789C,1236AB), of sizes ( 6 , 6 ) and ranks (4,4), respectively. By \4/5/2\ A get minor 13 7 14 10 1 2 8 3 = W4 Summands: Left= #2 1 - Right = G_y. A's: 12A 13B; A's: 26A 123; Y A's: 2A61 123A 132B; xi's: 4789 578C 459C; # 1 2 1 : (6,12) P Ref:#4 Y D u a l= #3 ii Bits': 59 61 14 35 5 3 /? = 3 DBC # of Cocys: (4,6,10,18,16,5,2,2,0,0); # of Cys: (3,9,11,11,17,10,1,1,0,0); # of R-psorbs=39; C-psorbs:l C B 6 4789 2 3A 5; # of Bases=400; Decomposition info: BTS (45789A,1236BC), of sizes ( 6 , 6 ) and ranks (4,4), respectively. By /4\9/2\ C get minor 4 13 15 91283 = W4 Summands: Left= #i§. Right= G-y. A 's: 1 2 c 13B 6 AC; /^'s: 578 459 56A 123; Y A's: A65C 123C 132B; m 's: 4789; # 1 2 2 : (6,12) P Ref; # 4 7 D u a l= # io9 Bits': 59 61 14 35 5 6 /? = 3 DBC # of Cocys: (5,5,9,19,15,6,3,1,0,0); # of Cys: (4,8,10,12,16,11,2,0,0,0); # of R-psorbs=39; C-psorbs:B 3 6 4789 A 2 1 5 C; # of Bases=376; Decomposition info: BTS (45789C,1236AB), of sizes ( 6 , 6 ) and ranks (4,4), respectively. By /4\ C /1\ B get minor 4 13 14 31289 = W4 Summands: Left= #ig. Right = G_y. A's: 13B 23C 49C 78C; /^'s: 578 16B 459 56A 123; Y A's: 875C 1B63 495C 132B 321C; ix's: 4789; #123: (6,12) P Ref:#47 Dual=#in Bits': 59 61 14 35 5 7 /? = 3 DBC # of Cocys: (3,6,14,16,12,9,2,0,1,0); # of Cys: (2,10,10,16,14,5,6,0,0,0); # of R-psorbs=25; C-psorbs:B 123C 4789A 5 6 ; # of Bases=440; Decomposition info: BTS (45789A,1236BC), of sizes ( 6 , 6 ) and ranks (4,4), respectively. By /4\9\l/3 get minor 4 15 13 11 2 8 1 3 = W4 244 Sum m ands: Leffc= TV^g. Right = Nij. A's: 13B 2BC; 578 459 56A; s: 4789 123C; # 1 2 4 : (6,12) P Ref: # 4 7 D u a l= #2 i 4 Elts': 59 61 14 35 5 19 /3 = 3 DEC # of Cocys: (1,10,12,16,14,5,4,0,1,0); # of Cys: (1,10,12,16,14,5,4,0,1,0); 7^ of R-psorbs=13; C-psorbs:B 13 2 456789AC; # of Bases=480; Decomposition info: FCBTS (4789,12356ABC) Summands: Left= Fj by /l\2/3/5\B . Right = #29 by /4. A 's: 13B; /j\'s: 123; Y A 's: 132B; x 's : 4789 56AC; #125: (6,12) P Ref: # 4 7 D u a l= # n9 E lts': 59 61 14 35 5 22 ^ = 3 DEC # of Cocys: (2,7,15,15,13,8,1,1,1,0); # of Cys: (1,11,11,15,15,4,5,1,0,0); ^ of R-psorbs=23; C-psorbs:B 6 1 2 3 45789AC; # of Bases=468; Decomposition info: BTS (45789C,1236AB), of sizes ( 6 , 6 ) and ranks (4,4), respectively. By \4 /5 /l\B get minor 13 14 731289 = W4 Summands: Left= #2 %. Right = G _ 7 - A's: 13B; A's: 16B 123; Y A's: 1B63 132B; x 's : 4789 578C 459C; #126: (6,12) PRef : # 4 7 D u a l= # io7 Elts': 59 61 14 35 5 24 /? = 3 DEC # of Cocys: (3,7,13,15,13,8,3,1,0,0); # of Cys: (3, 7,13,15,13,8,3,1,0,0); ^ of R-psorbs=55; C-psorbs:B C 35 8 29 7A 1 4 6 ;^ of Bases=418; Decomposition info: BTS (45789C,1236AB), of sizes ( 6 , 6 ) and ranks (4,4), respectively. By /4\ C /1\B get minor 4 13 14 31289 = W4 Summands: Left= #1 9 . R ight= G_j. A's: 13B 45C -7AC; A'a: 16B 459 123; Y A's: 1B63 459C 132B; # 1 2 7 : (6,12) P Ref: # 4 7 D u a l= #2 i 7 Elts': 59 61 14 35 5 36 ^8 = 3 DEC # of Cocys: (2,8,14,14,14, 7,2, 2,0,0); # of Cys: (2,8,14,14,14, 7, 2 , 2 ,0,0); # of R-psorbs=2 2 ; C-psorbs:16BC 2A 3 4789 5; # of Bases=456; Decomposition info: BTS (45789A,1236BC), of sizes ( 6 , 6 ) and ranks (4,4), respectively. By /4 \9 \l/2 get minor 4 13 15 9 2 8 3 10 = W4 Summands: Left= #ig. Right = # 1 7 . A's: 13B 36C; /^'s: 578 459; x 's : 4789 16BC; #128: (6,12) P Ref: # 4 7 Dual=#iig Elts': 59 61 14 35 5 37 /3 = 3 DEC # of Cocys: (4,5,11,19,15,6,1,1,1,0); # of Cys: (2,12,8,12,18, 7,4, 0,0,0); 245 # of R-psorbs=39; C-psorbs:B 1 3 A 2 4789 5 6 C; # of Bases=408; Decomposition info; BTS (45789C,1236AB), of sizes ( 6 , 6 ) and ranks (4,4), respectively. By /4 \9 /l\B get minor 4 13 14 10 1 2 8 9 = W4 Summands: Lefts N2Q. R ig h ts G-j. A's: 13B 6 BC; 4 \'s: 578 16B 459 123; Y A's: 1B63 B61C 132B; m 's: 4789; Y 1 2 9 : (6,12) P Ref:iV4 7 Dual^A^iss Bits': 59 61 14 35 5 38 /3 = 3 D EC # of Cocys: (3,6,14,16,12, 9,2 , 0 ,1,0); # of Cys: (2,10,10,16,14,5,6,0,0, 0); # of R-psorbs=39; C-psorbs:B AC 24789 1 3 5 Q; ^ oi Bases=436; Decomposition info: BTS (45789A,1236BC), of sizes ( 6 , 6 ) and ranks (4,4), respectively. By /4\9\l/2 get minor 4 13 15 9 2 8 3 10 S W4 Summands: Lefts N\g. R ig h ts Nig. A's: 13B -ABC; /^'s: 578 459 123; Y A's: 132B; x 's: 4789; #130: (6,12) P Ref: # 4 7 D u a l= #3 io B its': 59 61 14 35 5 48 ^ = 3 DEC # of Cocys: (2,8,14,14,14,7,2,2,0,0); # of Cys: (2,8,14,14,14,7,2,2,0,0); # of R-psorbs—23; C-psorbs:16 BC 4789 35 2A; # of Bases=448; Decomposition info: FCBTS (4789,12356ABC) Summands: Lefts Fj by /l\2 /3 / A \ B. Rights G _ 3 by /4. A's: 13B 56C; 4 \'s: 56A 123; Y A's: 56AC 132B; x 's: 4789; # 1 3 1 : (6 ,1 2 ) P Ref: # 4 7 D u a l= #2 ii E lts': 59 61 14 35 5 51 /? = 3 DEC # of Cocys: (3,9,11,11,17,10,1,1,0,0); # of Cys: (4,6,10,18,16,5,2,2,0,0); # of R-psorbs=39; C-psorbs:A B C 35 1 2 4789 6 ; # of Bases=400; Decomposition info: BTS (45789C,1236AB), of sizes ( 6 , 6 ) and ranks (4,4), respectively. By \4/7/l\B get minor 4 15 13 2 14 1 8 6 S W4 Summands: Lefts #ig. Rights G_ 7 . A's: 13B 47C 5AC 89C; ^s: 16B 56A 123; Y A's: 1B63 5A6C 132B; x 's : 4789; # 132: (6,12) P Ref:#47 D u al= #iog E lts': 59 61 14 35 5 53 /? = 3 DEC # of Cocys: (3,7,13,15,13,8,3,1,0,0); # ofCys: (3,7,13,15,13,8,3,1,0,0); # of R-psorbs=35; C-psorbs:B C 3 25A 6 4789 1; 41^ ofBases=420; Decompo.sition info: BTS (45789C,1236AB), of sizes ( 6 , 6 ) and ranks (4,4), respectively. By \4/5/l\B get minor 13 14 7 10 1 2 8 9 S W4 Summands: Lefts #1 7 . R ig h ts G- 7 . A's: 13B 48C 79C; z^'s: 16B 56A 123; Y A's: 1B63 132B; 1x 1' s: 4789; 2 4 6 #133: (6,12) P Ref: # 4 8 D ual=iV i3 3 Elts'; 27 61 46 35 5 7 /? = 3 DEC # of Cocys: (2,6,18,16,6,9,6,0,0,0); # of Cys: (2,6,18,16,6,9,6,0,0,0); # of R-psorbs=24; C-psorbs:5 B 123C 4789 6 A; # of Bases=480; Decomposition info: BTS (1236ABC,45789), of sizes (7,5) and ranks (4,4), respectively. By \l\2 /3 /4 get minor 8111347 13 10 S IV4 Summands: Left= #3 0 - Right = # 7 . A's: 13B 2BC; /|\'a: 578 459; x i's : 4789 123C; # 1 3 4 : (6,12) P Ref: # 4 8 D u a l= #3 i 4 Elts': 27 61 46 35 5 11 /3 = 3 DEC # of Cocys: (2, 7,16,15,10, 8 ,4 , 1 ,0,0); # of Cys: ( 2 , 7,16,15,10,8,4,1,0,0); ^ of R-psorbs=33; C-psorbs:35 69 28 BC 4A 17; 7^ of Bases=461; Decomposition info: BTS (45789C,1236AB), of sizes ( 6 , 6 ) and ranks (4,4), respectively. By \4 /8 /l\B get minor 47531 14 89 = W4 Summands: Left= Njg. Right = # 1 9 . A 's: 13B 57C; /|\'s: 578 123; Y A 's: 578C 132B; # 1 3 5 : ( 6 , 1 2 ) P Ref: # 4 8 D u a l= #2 i 9 Elts': 27 61 46 35 5 19 /? = 3 DEC # of Cocys: (2,10,10,16,14,5,6,0,0,0); # of Cys: (3,6,14,16,12,9,2 , 0,1, 0); # of R-psorbs=39; C-psorbs:4789B 56 2 3 1 A C; 7^ of Bases=436; Decomposition info: BTS (45789C,1236AB), of sizes ( 6 , 6 ) and ranks (4,4), respectively. By \4/5/6\ A get minor 11 13 14 3 1 2 4 5 = W4 Summands: Left= #1 7 . Right = Nig. A's: 13B 47C 89C; 4 's: 123 -256; Y A's: 132B; xi's: 4789; # 1 3 6 : ( 6 , 1 2 ) P Ref: # 4 8 D u a l= #3 ic Elts': 27 61 46 35 5 38 ^ = 3 DEC # of Cocys: (4,6,12,16,12,9,4,0,0,0); # of Cys: (4,6,12,16,12,9,4,0,0,0); ^ of R-psorbs=39; C-psorbs:134789 2 B 6 A C 5; 7^ of Bases=404; Decomposition info: BTS (45789C,1236AB), of sizes ( 6 , 6 ) and ranks (4,4), respectively. By /4\ C /1\B get minor 4 5 14 11 1289 = W4 Summands: Left= N\g. Right = # 1 9 . A's: 13B 49C 78C -ABC; /|\'s: 578 459 123 -256; Y A's: 875C 495C 132B; txi's: 4789; # 1 3 7 : (6,12) P Ref: # 4 8 D u a l= # ii6 Elts': 27 61 46 35 5 53 /? = 3 DEC # of Cocys: (2,8,16,12,10,11,4,0,0,0); # of Cys: (3,5,15,19,9,6,5,1,0,0); # of R-psorbs=39; C-psorbs:l 2 C 4789 3 5 B A 6 ; 7^ of Bases=448; 247 Decomposition info: BTS (45789C,1236AB), of sizes (6,6) and ranks (4,4), respectively. By \4 /5 /l\B get minor 5 14 15 10 1 2 8 9 = W4 Summands: Lefts jVij. R ig h ts Nig. A's: 13B 48C 79C; 123 -256; Y A's: 132B; xi's: 4789; N i 3 g: (6,12) P Ref:AT 4 8 D u a l= N i3 g Bits': 27 61 46 35 5 54 ^ = 3 DEC # of Cocys: (1,7,19,15,7,8,5,1,0,0); # of Cys: (1, 7,19,15,7, 8,5,1,0,0); ^ of R-psorbs=23; C-psorbs:3 2 45789C B 1 6 A; ^ of Bases—512; Decomposition info: BTS (45789C,1236AB), of sizes (6,6) and ranks (4,4), respectively. By \4 /5 /l\B get minor 5 14 15 11 1 2 8 9 S W4 Summands: Lefts N21. R ig h ts N\g. A's: 13B; 123; Y A 's: 132B; tx's: 4789 578C 459C; Nug: (6,12) P Ref:#4 g DuaI=A^i63 Bits': 27 29 14 35 37 3 ,9 = 3 DBC # of Cocys: (4,4,12,22,12,3,4,2,0,0); # of Cys: (2,10,14,8,14,13,2, 0, 0, 0); ^ of R-psorbs=23; C-psorbs:C 16 5 2A 34789B; ^ of Bases=420; Decomposition info: BTS (45789C,1236AB), of sizes (6,6) and ranks (4,4), respectively. By /4 \9 /l\2 get minor 4561289 10 S W4 Summands: Lefts jVoQ- Rights G_g. A 's: 1 2 c 6AC; A's: 578 6AB 459 123; Y A 's: 6ABC 123C; tx' s: 4789; ®'s : 6AB123; iVi4 o: (6,12) P Ref: Y 4 9 D u a l= Y i6 5 Bits': 27 29 14 35 37 6 /) = 3 DBC # of Cocys: (5,5,9,19,15,6,3,1,0,0); # of Cys: (4,8 ,1 0 ,1 2 ,16,11,2,0,0,0); ^ of R-psorbs=23; C-psorbs:16 4789 23AB 5 C; ^ of Bases=384; Decomposition info: BTS (45789C,1236AB), of sizes ( 6 , 6 ) and ranks (4,4), respectively. By /4\ C /1\2 get minor 4563289 1 0 S W4 Summands: LeftS Ni^. R ig h ts G _ 5 . A's: 23C 49C 78C ABC; /^'s: 578 6 AB 459 123 -156; Y A's: 875C BA 6 C 495C 321C; m 's: 4789; ©'s : 6AB123; Y i4i: (6,12) P Ref: # 4 9 Dual-Yiee Bits': 27 29 14 35 37 7 /? = 3 DBC # of Cocys: (3,4,14,22,12,3,2,2,1,0); # of Cys: (0,14,12, 8,16,9,4,0,0,0); # of R-psorbs=21; C-psorbs:5 123C 4789 6 AB; # of Bases=472; Decomposition info: BTS (1236ABC,45789), of sizes (7,5) and ranks (4,4), respectively. By \l/2\3/4 get minor 891134572S W4 248 Summands: Left= A^4 4 . R ig h ts Nj. V s: 578 6 AB 459; m 's: 4789 123C; #142: (6,12) P Ref: # 4 9 D u a l= #5 ic Bits': 27 29 14 35 37 11 /I - 3 DEC # of Cocys: (3, 5,14,19,12, 6, 2 , 1 , 1 , 0); # of Cys: (1,12,11,12,15, 7, 5,0,0,0); # of R-psorbs=35; C-psorbs:5 8 49 16 7 23AB C; # of Bases=447; Decomposition info: BTS (45789C,1236AB), of sizes ( 6 , 6 ) and ranks (4,4), respectively. By \4 /8 /l\2 get minor 4753689 14 = W4 Summands: Left= #1 9 . Right = (S'_ 5 . A 's: 57C; /\!s: 578 6 AB 123; Y A's: 578C; ®'s : 6AB123; # 1 4 3 : (6,12) P Ref; # 4 9 D u a l= # n3 Bits': 27 29 14 35 37 19 ^ = 3 DBC # of Cocys: (3,7,11,17,17,4,1,3,0,0); # of Cys: (2,10,12,10,18,9,0,2,0,0); # of R-psorbs=27; C-psorbs:C 16 2A 3B 4789 5; # of Bases=432; Decomposition info: BTS (45789C,1236AB), of sizes ( 6 , 6 ) and ranks (4,4), respectively. By \4/5/l\2 get minor 5671289 10 = W4 Summands: Left= A'^1 7 . Right = G_ 5 . A's: 47C 89C; Vs: 6AB 123 -156; ixi's: 4789; ©'s : 6AB123; #144: (6,12) P Ref: # 4 9 D u a l= #7 i 3 Bits': 27 29 14 35 37 22 ,9 = 3 DBC # of Cocys: (2,6,14,20,14,1,2,4,0,0); # of Cys: (0,12,16,6,16,11,0,2,0,0); ^ of R-psorbs—14; C-psorbs:16 23AB 45789C; ^ of Bases=492; Decomposition info: BTS (45789C,1236AB), of sizes ( 6 , 6 ) and ranks (4,4), respectively. By \4 /5 /l\2 get minor 5673289 10 = W 4 Summands: Left= #2 1 - Right = G_, 5 . Va: 6 AB 123; tx's: 4789 578C 459C; ©'s : 6AB123; # 1 4 5 : (6,12) P Ref: # 5 0 D u a l= :#6 i 0 Bits': 59 29 46 35 37 5 = 3 DEC # of Cocys: (4,4,12,22,12,3,4,2, 0,0); # of Cys: (2,10,14,8,14,13,2, 0,0,0); # of R-psorbs=24; C-psorbs:136B 2 4789 5 A C; ^ of Bases=424; Decomposition info: BTS (45789C,1236AB), of sizes ( 6 , 6 ) and ranks (4,4), respectively. By /4\9/l\3 get minor 4 13 62189 10 = W4 Summands: Left= #2 9 . Right = #i§. A 's: 13C 6 BC; Va: 578 26B 459 123; Y A's: B62C 132C; m 's: 4789 136B; 249 7Vl46: (6,12) P RePiVso Dual=iVi8 4 Bits': 59 29 46 35 37 7 ^0 = 3 DEC # of Cocys: (2,4,18,22,6,3,6,2,0,0); # of Cys: (0,10,20,8,8,13,4,0,0,0); # of R-psorbs=14; C-psorbs:5 1236BC 4789 A; # of Bases—512; Decomposition info: BTS (45789A,1236BC), of sizes (6,6) and ranks (4,4), respectively. By /4\9\l/2 get minor 4 13 7928 11 3 = IT4 Summands: Left= A^2 0 - Rights A^ 2 1 - Vs: 578 459; tx's: 4789 26BC 123C 136B; iVi4 7 : ( 6 ,1 2 ) P Ref:iV5 i B u a l^N ^j Bits': 27 45 62 357/1 = 3 DEC # of Cocys: (4,8 , 8 ,14,20,7,0,2, 0,0); # of Cys: (4 ,8 , 8 ,14,20,7,0,2, 0,0); # of R-psorbs=16; C-psorbs:123C 4789 AB 56; # of Bases=392; Decomposition info: BTS (1236ABC,45789), of sizes (7,5) and ranks (4,4), respectively. By \ l / 2 \ C /4 get minor 281345 11 14 SIP 4 Summands: Lefts A 3 3 . R ig h ts Nj. A 's: 12A 13B 2BC 3AC; /^'s: 579 689 458 467; tx's: 4789 123C; # 1 4 8 : ( 6 ,1 2 ) P R ef:#5 i D u a l= #4 ig Bits': 27 45 62 3 5 11 /? = 3 DBC # of Cocys: (4,7,10,15,16,8 , 2 ,1 ,0 ,0); # of Cys: (4,7,10,15,16,8 , 2 ,1 ,0 ,0); # of R-psorbs=63; C-psorbs : 6 1 7B 9 4 25 8 3 A C; # of Bases=385; Decomposition info: BTS (45789C,1236AB), of sizes ( 6 , 6 ) and ranks (4,4), respectively. By \4/8/2\ A get minor 4 13 15 9 11 1 8 10 S I/P4 Summands: Lefts #1 9 . Rights G-y. A's; 12A 13B -4AC 57C; V s: 579 -689 36B 123; Y A's: 579C 3B61 123A 132B; # 1 4 9 : (6,12) P Ref: # 5 1 D u a l= #4 i 9 Bits': 27 45 62 3 5 17 /3 = 3 DBC # of Cocys: (3,10,6,16,20,5,2,0,1,0); # of Cys: (3,10,6,16,20,5,2,0,1,0); # of R-psorbs=23; C-psorbs:l B 25AC 3 4789 6 ; ^ of Bases=408; Decomposition info: BTS (46789B,1235AC), of sizes ( 6 , 6 ) and ranks (4,4), respectively. By /4\7\l/2 get minor 8 11 14 32415S W4 Summands; Lefts #ig. Rights #ig. A's: 12A 13B 15C; V^: 689 36B 467; Y A's: 3B61; m 's: 4789 25AC; # 1 5 0 : (6,12) P Ref: # 5 1 D u a l= #5 io Bits': 27 45 62 3 5 19 ^ = 3 DBC # of Cocys: (5,7,7,15,19,8,1,1,0,0); # of Cys: (5,7,7,15,19,8,1,1,0,0); # of R-psorbs=39; C-psorbs:A 1 C 45789B 2 3 6 ; ^ of Bases=352; Decomposition info: BTS (46789C,1235AB), of sizes ( 6 , 6 ) and ranks (4,4), 250 respectively. By /4\ C /2\ A get minor 85 11 14 1243 = W4 Summands: Left= Nig. Right = G-g. A's: 12A 13B 47C 5AC 89C; 689 25A 36B 467 123; Y A's: 986C 2A51 A52C 3B61 476C 123A 132B; x,' s: 4789; Y i 5 i: (6,12) P Ref:Y 5 i D u aI= Y i5 i Elts': 27 45 62 3 5 51 /? = 3 D EC # of Cocys: (2,11,7,15,21,4,1,1,1,0); # of Cys: (2,11,7,15,21,4,1,1,1,0); # of R-psorbs=23; C-psorbs:l A 3 2 46789C 5 B; ^ of Bases=432; Decomposition info: BTS (46789C,1235AB), of sizes ( 6 , 6 ) and ranks (4,4), respectively. By \4/6/2\ A get minor 13 7 14 91283 = W4 Summands: Left= #9 %. Right = G-g. A's: 12A 13B; ^s: 25A 123; Y A's: 2A51 123A 132B; m 's: 4789 689C 467C; Y 152: (6,12) P Ref:Y5i D u a l= Y n2 Elts': 27 45 62 3 5 54 ^ = 3 DEC # of Cocys: (3,9,11,11,17,10,1,1,0,0); # of Cys: (4,6,10,18,16,5,2,2,0,0); # of R-psorbs=25; C-psorbs:AB 1 56 23 4789 C; of Bases=396; Decomposition info: BTS (46789C,1235AB), of sizes ( 6 , 6 ) and ranks (4,4), respectively. By \4/6/2\ A get minor 13 7 14 10 1 2 8 3 = W4 Summands: Left= Nij. Right= G_y. A's: 12A 13B 49C 78C; A's: 25A 36B 123; Y A's: 2A51 3B61 123A 132B; m 's: 4789; Y 153: (6,12) P Ref:Y52 D u a l= Y i53 Elts': 59 13 30 3 37 5 /? = 3 DEC # of Cocys: (5,6,9,16,15,9,3,0,0,0); # of Cys: (5,6,9,16,15,9,3,0,0,0); ^ of R-psorbs=39; C-psorbs:368 IB 49A 27 5 C; of Bases=361; Decomposition info: BTS (5679BC,12348A), of sizes ( 6 , 6 ) and ranks (4,4), respectively. By /5\9/4\8 get minor 8 11 13 51243 = W4 Summands: Left= G-j. Right = N\g. A 's: 12A 13C 48C 6 BC 79B; /[!s: 579 67B 25A 458 123; Y A's: 975B 6B7C B769 2A51 485C 123A 132C; A^154: (6,12) P Ref:Y52 Dual=A^i6i Elts': 59 13 30 3 37 6 /? = 3 DEC # of Cocys: (4,5,12,19,12,6,4,1,0,0); # of Cys: (3,8,13,12,13,11,3,0,0,0); # of R-psorbs=47; C-psorbs:5 2 7 9 6 AC 13B 48; ^ of Bases=403; Decomposition info: BTS (46789B,1235AC), of sizes ( 6 , 6 ) and ranks (4,4), respectively. By /4 \8 /l\ A get minor 8 13 7 10 1243 = W4 251 Summands: Left= TVig. Right= G--j. A's: 12A 23C 79B; /^s: 579 67B 458 123; YA'a: 975B B769 123A 321C; A^lSS: (6,12) P Ref:A^ 5 2 D u a l-iV n4 Elts': 59 13 30 3 37 7 /3 = 3 D E C # of Cocys: (3,7,13,15,13,8,3,1,0,0); # of Cys: (3,7,13,15,13,8,3,1,0,0); # of R-psorbs=35; C-psorbs:9 A 5 46 123C B 7 8 ; of Bases=424; Decomposition info: BTS (46789B,1235AC), of sizes ( 6 , 6 ) and ranks (4,4), respectively. By /4\8\l/2 get minor 8 13 6 11 2413 = W4 Summands: Left= Nig. R ig h ts Nn- A's: 12A 3AC 79B; J[!s: 579 67B 458; YA 'a: 975B B769; x 's : 123C; A^1 5 6 : (6,12) P Ref;A 5 2 D uaI=iV i4 2 Elts': 59 13 30 3 37 20 /? = 3 D E C # of Cocys: (1,12,11,12,15, 7,5, 0, 0,0); # of Cys: (3, 5,14,19,12,6 ,2,1,1,0); ^ of R-psorbs=27; C-psorbs:9 7 13 6 25AC B 48; # of Bases=447; Decomposition info: BTS (46789B,1235AC), of sizes ( 6 , 6 ) and ranks (4,4), respectively. By \4/6\l/2 get minor 13 7 14 3 2 8 1 10 = W4 Summands: Left= Yjg. Right = G_ 4 . A'a: 12A 35C 79B; 4 \'a: 67B; Y A 'a: B769; ©'a : 12A35C; Y 157: (6,12) P Ref:iV52 Dual=iVi57 Elts': 59 13 30 3 37 21 /3 = 3 DEC # of Cocys: (2,8,14,14,14,7,2,2,0,0); # of Cys: (2,8,14,14,14,7,2,2,0,0); # of R-psorbs=23; C-psorbs:9A 27 458C IB 36; # of Bases=452; Decomposition info: BTS (5679BC,12348A), of sizes ( 6 , 6 ) and ranks (4,4), respectively. By \5/6/4\8 get minor 11 14 5 13 1 2 4 3 = W4 Summands: Left= Yjg. Right = N\g. A's: 12A 79B; /|\'a: 67B 123; Y A 'a: B769 123A; m' a: 458C; Nigg: (6,12) P Ref:Ys2 D ual=iV i5 8 Elts': 59 13 30 3 37 24 /? = 3 A C T # of Cocys: (3,6,15,16,9,9,5,0,0,0); # of Cys: (3,6,15,16,9,9,5,0,0,0); ^ of R-psorbs=19; C-psorbs:257 9AC 368 14B; ^ of Bases=421; A'a: 12A 45C 79B; /^'s: 67B 458 123; YA'a: B769 458C 123A; A^isg: ( 6 , 1 2 ) P Ref: # 5 2 D u a l= #5 ig Elts': 59 13 30 3 37 29 /? = 3 ACT # of Cocys: (4,6,12,16,12, 9,4, 0, 0,0); # of Cys: (4,6,12,16,12,9,4,0,0,0); ^ of R-psorbs=19; C-psorbs:346 9AC 257 18B; # of Bases=394; A 'a: 12A 58C 79B -9AC; /|\'a: 67B 458 123 -346; YA'a: B769 854C 123A; 2 5 2 7Vl6Q: (6,12) P Ref:AT5 3 Dual=A^i4 5 Elts': 59 45 30 3 37 7 ^ = 3 DEC # of Cocys: (2,10,14,8,14,13,2,0,0,0);# of Cys: (4,4,12,22,12,3,4,2,0,0); # of R-psorbs=24; C-psorbs:B 5 4789 123C A 6; # of Bases=424; Decomposition info: BTS (46789B,1235AC), of sizes (6,6) and ranks (4,4), respectively. By /4\B \l/2 get minor 8 13 11 62413 = IV4 Summands: Left= Nig. Right= A’n- A's: 12A 3AC 48B 79B; /f^'s: 579 458; Y A's: 975B 485B; tx's: 4789 123C; NiQi: (6,12) P Ref:iV5 3 D u a l= Y i5 4 Elts': 59 45 30 3 37 14 /? = 3 DEC # of Cocys: (3,8,13,12,13,11,3,0,0,0); # of Cys: (4,5,12,19,12,6,4,1,0,0); # of R-psorbs=39; C-psorbs:7 B C 25A 48 36 1 9; # of Bases=403; Decomposition info: BTS (45789C,1236AB), of sizes (6,6) and ranks (4,4), respectively. By \4 /8 /l\ A get minor 4 15 13 9 1 10 8 2 = W4 Summands: Left= Nig. Right = A's: 12A 48B 59C 79B; /[!s: 579 16A 123; Y A 's: 597C 975B 1A62 123A; Y 1 6 2 : (6,12) P Ref:7V53 Duai^N ny Elts': 59 45 30 3 37 17 /? = 3 DEC # of Cocys: (0,18,0,24,0,21,0,0,0,0); # of Cys: (4,3,12,24,12,3,4,0,0,1); # of R-psorbs=9; C-psorbs:lB 245789AC 36; # of Bases=440; Decomposition info: BTS (46789B,1235AC), of sizes (6,6) and ranks (4,4), respectively. By \4 /6 \l/2 get minor 13 7 14 32819 = W4 Summands: Left= N n. Right = Nig. A's: 12A 15C 48B 79B; tx 's: 4789 25AC; NiQg: (6,12) P Ref:Y 5 3 Bnal=Nigc, Elts': 59 45 30 3 37 20 /? = 3 DEC # of Cocys: (2,10,14,8,14,13,2,0,0,0); # of Cys: (4,4,12,22,12,3,4,2,0,0); # of R-psorbs=25; C-psorbs:13 24789C 5A 6 B; # of Bases=420; Decomposition info: BTS (46789B,1235AC), of sizes (6,6) and ranks (4,4), respectively. By \4 /6 \l/A get minor 12 7 15 3 1 2 8 10 = W4 Summands: Left= Nn- Right= &'_ 4 . A's: 12A 35C 48B 79B; /|\'s: 16A 356; Y A's: 1A62 356C; m ' s : 4789; ®'s 12A35C; Y 1 6 4 : (6,12) P Ref:Y 5 3 B u a l^N m Elts': 59 45 30 3 37 22 ^ = 3 DEC # of Cocys: (4,8,10,12,16,11,2,0,0,0); # of Cys: (5,5,9,19,15,6,3,1,0,0); # of R-psorbs=25; C-psorbs:BC 56 12 4789 3 A; # of Bases=380; 253 Decomposition info: BTS (45789C,1236AB), of sizes ( 6 , 6 ) and ranks (4,4), respectively. By \4/5/6\B get minor 11 13 14 6 1 2 4 3 = W 4 Summands: Left= Right = G-y. A'a: 1 2 A 48B 49C 78C 79B; 16A 25A 123 356; Y A 'a: 1A62 2A51 123A; IX]' s: 4789; Y 165: (6,12) P Ref:iV53 D ual=N i4o Bits': 59 45 30 3 37 36 /3 = 3 DEC # of Cocys: (4,8,10,12,16,11,2,0,0,0); # of Cys: (5,5,9,19,15,6,3,1,0,0); ^ of R-psorbs=18; C-psorbs:lC 4789 236A 5 B; # of Bases=384; Decomposition info: BTS (45789B,1236AC), of sizes ( 6 , 6 ) and ranks (4,4), respectively. By /4\B \l/2 get minor 4 13 11 6 2 8 1 10 = W4 Summands: Left= Yjg- Right = G_ 4 . A'a: 1 2 A -IBC 36C 48B 79B; /|\'a: 579 25A 458 356; Y A'a: 975B 2A51 485B 365C; m' a: 4789; ©'a : 12A36C; Niqq: (6,12) P Ref:Y53 D ual=.Y i4i Bits': 59 45 30 3 37 52 /3 = 3 DEC # of Cocys: (0,14,12, 8,16, 9,4, 0 ,0, 0); # of Cys: (3,4,14,22,12,3,2,2,1,0); ^ of R-psorbs=16; C-psorbs:4789 1 2 A 356C B; of Bases=472; Decomposition info: BTS (12356AC,4789B), of sizes (7,5) and ranks (5,3), respectively. By \l/2 /3 \4 get minor 4 8 1 12 15 11 6 9 = W4 Summands: Left= Ygg. Right = Y 5 . A'a: 1 2 A 48B 79B; txi' a: 4789 356C; N 167: (6,12) P Ref:Y 5 3 D u a l= Y n o Bits': 59 45 30 3 37 53 /? = 3 DEC # of Cocys: (2,10,14,8,14,13,2, 0,0,0); # of Cys: (4,4,12,22,12,3,4,2,0,0); # of R-p.sorbs=15; C-psorbs:45789C 2A 1 B 36; 4^ of Bases=428; Decomposition info: BTS (45789BC,1236A), of sizes (7,5) and ranks (4,4), respectively. By \4/5\ C /3 get minor 15 13 691283 = W4 Summands: Left= N^Q- Right = G_g. A 'a: 1 2 A 48B 5BC 79B; 4 v'a: 16A 123; YA'a: 1A62 123A; x ' a: 4789 579C 458C; N 168 : (6,12) P Ref:/\^53 DuaWNno Bits': 59 45 30 3 37 54 /? = 3 DEC # of Cocys: (2,8,16,12,10,11,4,0,0,0); # of Cys: (3,5,15,19,9,6,5,1,0,0); 44 of R-psorbs=25; C-psorbs:lA 4789 35 6 C 2 B; of Bases=448; Decomposition info: BTS (46789BC,1235A), of sizes (7,5) and ranks (4,4), 254 respectively. By \4/6\ B /3 get minor 15 5 14 10 1 2 8 3 = W 4 Summands: Left= N^q. Right= G_g. A's: 12A 48B 79B; 25A 123; YA's: 2A51 123A; xi's: 4789; Nieg: (6,12) P Rei:N^4 Dual^A^igg Bits': 59 13 62 3 37 5 /? = 3 DEC # of Cocys: (5,7,7,15,19,8,1,1,0,0); # of Cys: (5,7,7,15,19,8,1,1,0,0); # of R-psorbs=23; C-psorbs:l 3 A 46789B C 2 5; # of Bases=368; Decomposition info: BTS (5679BC,12348A), of sizes ( 6 , 6 ) and ranks (4,4), respectively. By /5\6/4\8 get minor 11 14 13 5 1 2 4 3 = W4 Summands: Left= N\g. Right = N\g. A's: 12A 13C 480 6 BC 790; /t!s: 579 25A 458 56B 123; YA's: 9750 2A51 4850 B650 123A 1320; xi's: 4789 679B 468B; Y i 7 o: (6,12) P Ref:#M DuaI=Yi6 7 Bits': 59 13 62 3 37 6 /? = 3 DBO # of Oocys: (4,4,1 2 , 2 2 ,12,3,4,2, 0,0); # of Oys: (2,10,14,8,14,13,2,0,0, 0); # of R-psorbs—19; 0-psorbs:2 AO 5 13 46789B; ^ of Bases=428; Decomposition info: BTS (1236ABO,45789), of sizes (7,5) and ranks (4,4), respectively. By / 6 \ A \ B /4 get m inor 12468 11 5 14 = W4 Summands: Left= Y 3 5 . Right = Nj. A's: 12A 230; A's: 579 458 56B 123; YA's: 123A 3210; xi's: 4789 679B 468B; iVi7 i: (6,12) P Ref: # 5 4 D u a l= # ig2 Bits': 59 13 62 3 37 7 /? = 3 DBO # of Oocys: (3,6,14,16,12,9, 2 ,0,1,0); # of Oys: (2,10,10,16,14,5,6 ,0 ,0 ,0 ); ^ of R-psorbs=15; 0-psorbs:A 1230 46789B 5; ^ of Bases=448; Decomposition info: BTS (1236ABO,45789), of sizes (7,5) and ranks (4,4), respectively. By \ l / 6 \ B /4 get minor 24378 11 5 14 = W4 Summands: Left= #gg. Right = # 7 . A's: 12A 3A0; /^'s: 579 458 56B; x 's : 4789 679B 468B 1230; #172: (6,12) P Ref: # 5 4 D u a l= #7 i 6 Bits': 59 13 62 3 37 17 /) = 3 DBO # of Oocys: (0,16,0,30,0,15, 0,2,0,0); # of Oys: (2 , 7,14,16,14, 7, 2 ,0 ,0 ,1 ); # of R-psorbs=10; 0-psorbs:l 25AO 346789B; # of Bases=480; Decomposition info: BTS (46789B,1235AO), of sizes ( 6 , 6 ) and ranks (4,4), respectively. By \4 /6 \l/2 get minor 13 7 14 32819 = IT4 Summands: Left= #2 1 - R ight= #%g. A's: 12A 150; x 's : 4789 679B 25AO 468B; 255 N i 7 3 : (6,12) P Ref: # 5 4 D u a l= # M4 Bits': 59 13 62 3 37 20 /I = 3 DEC # of Cocys: (0,12,16,6,16,11,0,2,0,0); # of Cys: (2,6,14,20,14,1,2,4,0,0); # of R-psorbs=ll; C-psorbs:25AC 13 46789B; # of Bases=492; Decomposition info: BTS (46789B,1235AC), of sizes ( 6 , 6 ) and ranks (4,4), respectively. By \4 /6 \l/2 get minor 13 7 14 3 2 8 1 10 = W4 Summands: Left= #2 1 - Right = G_ 4 - A 's: 12A 35C; x 's : 4789 679B 468B; ©'s : 12A35C; A^174: (6,12) P Ref:#54 Dnal=Nij 4 Bits': 59 13 62 3 37 21 /3 = 3 DEC # of Cocys: (1,10,12,16,14,5,4,0,1,0); # of Cys: (1,10,12,16,14,5,4,0,1,0); # of R-psorbs=13; C-psorbs:A 3 456789BC 2 1; 7^ of Bases=488; Decomposition info: BTS (5679BC,12348A), of sizes ( 6 , 6 ) and ranks (4,4), respectively. By \5/6/4\8 get minor 11 14 5 13 1 2 4 3 = W4 Sum mands: Left= A'?!- Right = Nig. A's: 12A; /|\'s: 123; YA's: 123A; txi's: 4789 579C 679B 458C 468B 56BC; A 1 7 5 : (6,12) P Ref:A 5 5 Dual=Aisi Bits': 59 13 62 35 37 5 - 3 DEC # of Cocys: (5,6,8,16,18,9,0,0,1,0); # of Cys: (4,10,4,16,20,5,4,0,0,0); 7^ of R-psorbs—17; C-psorbs:A 1346789B C 25; # of Bases=376; Decomposition info: BTS (45789C,1236AB), of sizes ( 6 , 6 ) and ranks (4,4), respectively. By /4\ C \l / 2 get minor 4 13 3 14 2 8 9 11 = W4 Summands: Lefts Nig. R ig h ts Nig. A's: 13C 48C 6 BC 79C; 579 25A 26B 458 123; YA's: 975C B62C 485C 132C; m ' s : 4789 136B; A i7c: (6,12) P Ref:A 5 5 D u a l= A i7 2 Bits': 59 13 62 35 37 7 ^ = 3 DEC # of Cocys: (2,7,14,16,14,7,2,0,0,1); # of Cys: (0,16,0,30,0,15,0,2,0,0); # of R-p.sorbs=13; C-psorbs: 1236ABC 4789 5; # of Bases=480; Decomposition info; BTS (45789A,1236BC), of sizes ( 6 , 6 ) and ranks (4,4), respectively. By /4\8\l/2 get minor 4 13 14 9 2 8 11 3 S W4 Summands: Lefts Nig. R ig h ts A 2 1 . 4 \'s: 579 458; m 's: 4789 26BC 123C 136B; A]77: (6,12) P Ref:A55 D u a l= A i6 2 Bits': 59 13 62 35 37 38 /? = 3 DEC # of Cocys: (4,3,12,24,12,3,4,0,0,1);# of Cys: (0,18,0,24,0,21,0,0,0,0); # of R-psorbs=14; C-psorbs: 1346789B 25 AC; # of Bases=440; 256 Decomposition info: BTS (45789C,1236AB), of sizes ( 6 , 6 ) and ranks (4,4), respectively. By /4 \8 /l\3 get minor 4 13 15 11 1 8 9 10 = W4 Summands: Left= A^2 0 - Right = # 2 0 . y a: 579 26B 458 123; tx's: 4789 136B; #178: (6,12) P Ref: # 5 5 D u a l= #7 ig Bits': 59 13 62 35 37 49 /? = 3 DEC # of Cocys: (0,15,0,32,0,15,0,0,0,1); # of Cys: (0,15,0, 32, 0,15,0,0,0,1); # of R-psorbs=4;:ÿ)^ of C-psorbs=l; # of Bases=496; Decomposition info: FCBTS (4789,12356ABC) Summands: Lefts Fj by \l/2/3/6\B . Rights # 3 7 by /4. tx's: 4789 25AC 136B; # 1 7 9 : (6,12) P Ref: # 5 6 D u a l= #7 i 9 Bits': 27 45 62 35 37 3 ^ = 3 DEC # of Cocys: (4,6,12,16,12,9,4, 0,0,0); # of Cys: (4, 6,12,16,12, 9,4,0,0,0); # of R-psorbs=14; C-psorbs: 1246789A 35 BC; # of Bases=392; Decomposition info: BTS (45789C,1236AB), of sizes ( 6 , 6 ) and ranks (4,4), respectively. By \4 /5 /l\2 get minor 5 14 15 1 2 8 9 10 S W4 Summands: Lefts A^ig. Rights #ig. A 's: 1 2 c 48B 6 AC 79B; ^'s: 579 36A 458 123; YA's: 975B A63C 485B 123C; tx's: 4789 126A; # 1 8 0 : (6,12) P Ref: # 5 6 Dual=#igo Bits': 27 45 62 35 37 7 /? = 3 DEC # of Cocys: (2,8,14,14,14,7,2,2,0,0); # of Cys: (2,8,14,14,14, 7 ,2,2,0,0); # of R-psorbs=14; C-psorbs:4789 1236AC B 5; # of Bases=464; Decomposition info: BTS (45789B,1236AC), of sizes (6,6) and ranks (4,4), respectively. By /4\B \l/3 get minor 479 14 28 11 3 S W 4 Summands: LeftS #ig. Rights # 2 1 . A's: 48B 79B; 4\'s: 579 458; YA's: 975B 485B; x 's : 4789 36AC 123C 126A; #lgi: (6,12) P Ref: # 5 6 D u a l= #7 i 5 Bits': 27 45 62 35 37 33 = 3 DEC # of Cocys: (4,10,4,16,20,5,4,0,0,0); # of Cys: (5,6,8,16,18,9,0,0,1,0); # of R-psorbs=12; C-psorbs:BC 3 1246789A 5; of Bases=376; Decomposition info: BTS (45789B,1236AC), of sizes (6,6) and ranks (4,4), respectively. By /4 \B /1 \C get minor 4 5 10 15 1 2 8 9 S W4 Summands: Lefts #ig. Rights #ig. 257 A 's : 16C 2AC 3BC 48B 79B; A's: 579 25A 458 156; YA'a: 975B 2A5C 485B 165C; txi's; 4789 126A; Yi82: (6,12) P RePA^so Dnal=Nm B its': 27 45 62 35 37 53 /? = 3 DEC # of Cocys: (2,10,10,16,14,5,6,0,0,0); # of Cys: (3,6,14,16,12,9,2,0,1,0); # of R-psorbs=13; C-psorbs:45789C 126A 3 B; # of Bases=448; Decomposition info: BTS (45789BC,1236A), of sizes (7,5) and ranks (4,4), respectively. By \4/5\ C /I get m inor 5 14 15 10 1 2 8 9 = W4 Sum m ands: L eft= Ygg. R ight = N-j. A 's : 48B 5BC 79B; /^'s: 36A 123; x ' a: 4789 579C 458C 126A; Y 1 8 3 : (6,12) P Ref:Y 5 7 Dual^Yigg Bits': 59 45 14 51 5 43 /3 = 3 DEC # of Cocys: (3,7,13,15,13,8,3,1,0,0); # of Cys: (3,7,13,15,13,8,3,1,0,0); # of R-psorbs=31; C-psorbs:35AB 89 4C 26 1 7; # of Bases=418; Decomposition info: BTS (45789AC,1236B), of sizes (7,5) and ranks (4,4), respectively. By \4/5\ C /2 get minor 13 15 691283 = W4 Summands: Left= Y 3 9 . Right = G_g. A's: 13B 47A 57C; A's : 57A 16B 123; YA's: 57AC A754 1B63 132B; Y 184 : (6,12) P Reî:N^s D u a l= Y i4 6 Bits': 27 61 46 51 5 7 /? = 3 DEC # of Cocys: (0,10,20,8,8,13,4,0,0,0); # of Cys: (2,4,18,22,6,3,6,2,0,0); # of R-psorbs=14; C-psorbs:B 46789A 5 123C; # of Bases=512; Decomposition info: BTS (46789A,1235BC), of sizes ( 6 , 6 ) and ranks (4,4), respectively. By \4/6\l/3 get minor 15 13 6 11 2 8 1 3 = IV4 Summands: Left= A 2 1 . Right = Nij. A 's : 13B 2BC; tx's: 4789 689A 467A 123C; Y 185 : (6,12) P RePiVco D u a l= N i85 Bits': 43 29 62 51 3 7 7 ^ = 3 DEC # ofCocys: (0,6,24,16,0,9 ,8 , 0 ,0 ,0 ); ^ of Cys: (0,6,24,16,0,9,8,0,0,0); # of R-psorbs=5;# of C-psorbs=l; # of Bases=576; Decomposition info: BTS (45789A,1236BC), of sizes ( 6 , 6 ) and ranks (4,4), respectively. By \4/5\l/2 get minor 13 7 14 9 2 8 11 3 = W4 Summands: Left= #2 1 - R ight= # 2 1 - tx's: 4789 589A 26BC 457A 123C 136B; 258 N i 8 G: (6,12) P Ref:A^61 Dual=jVig6 Bits': 11 61 62 35 7 3 /? = 3 DEC # of Cocys: (5,10,0,16,26,5,0,0,1,0);# of Cys: (5,10,0,16,26,5,0,0,1,0); # of R-psorbs=7; C-psorbs: 12346789AB 5 C; # of Bases=368; Decomposition info: BTS (356ABC,124789), of sizes ( 6 , 6 ) and ranks (4,4), respectively. By \3/5/l\2 get minor 8 9 3 1 4 5 14 15 = # 3 Summands: Left= #25- Right= # 2 5 . A's: 1 2 c 3BC 47C 6 AC 89C; 589 35B 457 56A 125; Y A's: 985C 3B5C 475C A65C 125C; m ' s : 389B 4789 689A 1289 347B 36AB 467A 123B 1247 126A; #187: (6,12) P Ref:#6 i D u a l= # fg7 Bits': 11 61 62 35 7 19 ^ = 3 DEC # of Cocys: (0,15,0,32,0,15,0,0, 0,1); # of Cys: (0,15,0, 32,0,15,0,0,0,1); # of R-psorbs=4;# of C-psorbs=l; # of Bases=512; Decomposition info: BTS (356ABC,124789), of sizes ( 6 , 6 ) and ranks (4,4), respectively. By \3/5/l\2 get minor 893145 14 15 = # 3 Summands: Left= #2 5 • Right = # 2 5 . m ' s : 389B 4789 589C 689A 1289 347B 35BC 36AB 457C 467A 56AC 123B 1247 125C 126A; #188: (6,12) D R e f:# ii7 Bits': 27 13 22 7 35 37 /3 = 3 DEC 259 T able A .I . Branch-width distribution = 3 13 = A Total (3,7) 1 0 1 4,7) 1 0 1 (4,8) 2 0 2 (4,9) 2 0 2 (5,9) 2 0 2 (4,10) 2 1 3 (6,10) 2 1 3 4 , 1 1 ) 1 0 1 7,11) 1 0 1 (5,10) 10 1 11 (5,11) 18 0 18 (6,11) 18 0 18 (5,12) 20 0 20 (7 , 1 2 ) 20 0 20 ( 6 , 1 2 ) 85 0 85 Totals 185 3 188 List A .3. Obstacles to branch-width 3 (three) N il: (4,10) P RefrNs D ual^N n Elts': 11 13 14 3 5 9 = 4 N 1 4 : (6,10) D Ref:Nil Elts': 59 13 22 39 /? = 4 N 2 3 : (5,10) P Ref:N? Dnai^Ngg Elts': 27 13 22 7 25 ^ = 4 List A .4. Self dual matroids (38) N 3 : (4, 8 ) P Ref:N 2 Exts=2 DuaI=N3 Elts': 11 13 14 3 /? = 3 ACT N 4 : (4, 8 ) P Ref:N 2 Dual=N| Elts': 11 13 14 7 /? = 3 DEC Nig (5.10) P Ref:Ny Exts=6 Dual=Nig Elts': 27 13 22 7 5 /? = 3 DEC N i 9 (5.10) P RefN ? Exts=l Dual=Nig Elts': 27 13 22 7 9 /) = 3 ACT N 2 1 (5.10) P Ref:N? Exts=l Dual=N2 i Elts': 27 13 22 7 14 /3 = 3 DEC N 2 3 (5.10) P Ref:Ny Dual=N 23 Elts': 27 13 22 7 25 /3 = 4 N 2 4 (5.10) P Ref:Ny D u al= N2 4 Elts': 27 13 22 7 29 /? = 3 DEC N 2 5 (5.10) P Ref:Ng Exts=l Dual=N 2 5 Elts': 27 29 22 7 6 ^ = 3 DEC N 2 6 (5.10) P Ref:Ng D u aI= N2 6 Elts': 27 29 22 7 14 /? = 3 DEC N 1 0 5 : (6,12) P Ref:N 4 6 D u al= N io5 Elts': 59 29 46 3 5 7 /? = 3 DEC 26 0 A^lOG (6 12 P Ref:A^4(5 D u a l= # io 6 E lts' 59 29 46 3 5 13 /? = 3 DEC ■^115 (6 12 P Ref:A^46 D u a l = # i5 i E lts' 59 29 46 3 5 38 /3 = 3 DEC ■^120 (6 12 P R ef: # 4 6 D u a I = #2 io E lts' 59 29 46 3 5 54 /3 = 3 DEC ^ 124 (6 12 P R e f:# 4 7 D u a l= # 'i2 4 E lts' 59 61 14 35 5 19 = 3 DEC N \2 1 (6 12 P R e f:# 4 7 D u a l = # i 2 7 E lts' 59 61 14 35 5 36 = 3 DEC -^130 (6 12 P R ef: # 4 7 D ucJ = # i 3o E lts' 59 61 14 35 5 48 = 3 DEC •^133 (6 12 P Ref: #4 8 Dual=#i33 Elts' 27 61 46 35 5 7 /? = 3 DEC -^134 (6 12 P R ef: # 4 8D u a l = #34 i E lts' 27 61 46 35 5 11 /? = 3 DEC -^136 (6 12 P Ref: #4 8 D ual=#i36E lts' 27 61 46 35 5 38 ^ = 3 DEC -^138 (6 12 P Ref: #4 8 Dual=#i38E lts' 27 61 46 35 5 54 ^ = 3 DEC -^147 (6 12 P R ef: # 5 1 D u a l = #47 i E lts' 27 45 62 3 5 7 /3 3 DEC -^148 (6 12 P R ef: #5 1 D u a l = # i 4 8 E lts' 27 45 62 3 5 11 /3 = 3 DEC -^149 (6 12 P R ef: # 5 1 D u a l = # i 4 9 E lts' 27 45 62 3 5 17 /? = 3 DEC -^150 (6 12 P R ef: #51 D u a l = # i 5 o E lts' 27 45 62 3 5 19 /3 = 3 DEC Nl5l (6 12 P Ref: #51 D ual=#i5i E lts' 27 45 62 3 5 51 /? = 3 DEC ■^153 (6 12 P R e f:# 5 2 Dual=#i53 Elts' 59 13 30 3 37 5 /? = 3 DEC -^157 (6 12 P R ef: # 5 2 D u a l = # i 5 7 E lts' 59 13 30 3 37 21 /? = 3 DEC -^158 (6 12 P R e f:# 5 2 D u a l= # i5 8 E lts' 59 13 30 3 37 24 ,5 = 3 ACT •^159 (6 12 P R ef: # 5 2 D u a l = #59 i E lts' 59 13 30 3 37 29 /Î = 3 ACT -^169 (6 12 P R e f:# 5 4 D u a l = # i 6 9 E lts' 59 13 62 3 37 5 /3 = 3 DEC 7Vi 74 (6 12 P R e f:# 5 4 D u a l = # i 7 4 E lts' 59 13 62 3 37 21 /? = 3 DEC -^178 (6 12 P R e f:# 5 5 D u a l = # i 7 8 E lts' 59 13 62 35 37 49 /3 = 3 DEC -^179 (6 12 P R ef: # 5 6 D u al= # i7 9 E lts' 27 45 62 35 37 3 /? = 3 DEC -^180 (6 12 P R ef: # 5 6 D u a l = #80 i E lts' 27 45 62 35 37 7 ^ = 3 DEC -^183 (6 12 P R ef: A%7 D u a l = # i 8 3 E lts' 59 45 14 51 5 43 /9 = 3 DEC -^185 (6 12 P R ef: # 6 0 D u a l = # i 8 5 E lts' 43 29 62 51 37 7 ^ = 3 DEC -^186 (6 12 P R ef: # 6 1 D u a l = # i 8 6 E lts' 11 61 62 35 7 3 /? = 3 DEC iVl87 (6 12 P R ef: # 6 1 D u al= # fg 7 E lts' 11 61 62 35 7 19 /3 = 3 DEC 261 Table A .2. 4-structure distribution (for matroids of branch-width 3) Y A 's 0 (in 0 M’s) 7 (in 1 M’s) n 2 M ’s n 2 M s n 2 M s n 2 M s 11 2 M ’s n 1 M ’s n 1 M s n 9 M s 37 (in 16 M ’sin 11 M ’s 37 (in 16 M ’sin 30 m 11 M ’s 37 in 16 M ’s 33 in 12 M ’s 34 in 18 M ’s 33 (in 12 M s 34 in 18 M ’s 165 (in 72 M ’s177 in 62 M ’s 165 (in 72 M ’s177 Totals 332 (in 119 M ’s 388 (in 163 M’s) Table A 3. 3-, and 6-structure distribution (for matroids of branch-width 3) 0 (in 0 M s (in 1 M s 0 (in 0 M s7 in 0 M sin 1 M sin 0 M sin 0 (in 0 M s in 1 M s in 1 M s 0 (in 0 M s in 2 M ’sin 1 M s in 2 M ’sin 10 (in 2 M ’s in 1 M s 0 (in 0 M ’s 1 (in 1 M s in 2 M s 0 in 0 M ’s 18 (in 2 M s in 1 M ’s 0 (in 0 M s 0 (in 0 M s in 1 M ’s 0 (in 0 M s 13 (in 1 M s in 0 M s 0 (in 0 M s 21 in 8 M ’s in 8 M ’s 2 in 2 M ’s 22 n 13 M’s in 17 M s 2 in 2 M ’s 81 in 17 M ’s in 13 M s 2 in 2 M ’s 17 in 12 M ’s in 19 M*s 2 (in 2 M ’s 142 (in 19 M’s in 12 M ’s) 2 (in 2 M ’s 233 in 77 M’s in 77 M 's) 10 (in 10 M*s Totals 569 (in 155 M ’s) 569 (in 155 M ’s 20 in 20 M’s 26 2 List A.5. Matroids that have a 6-structure (20) N22 5.10) P RehiVy Dual=A 27 Elts': 27 13 22 7 24 /? = 3 DEC N 27 5.10) D Ref:#22 Exts = 2 Elts': 11 13 14 19 21 ^ 3 DEC ^31 5.11) P Ref:#17 D u a l= #4g Elts': 27 13 22 7 3 24 ^ = 3 DEC A^44 5.11) P Ref:#27 Exts = 2 Dual= # 6 2 Elts': 11 13 14 19 21 3 ,3 = 3 DEC -^49 6.11) D Ref:#31 Exts = 6 Elts': 27 29 14 35 37 ^ = 3 DEC ■^62 6.11) D Ref:#44 Elts': 59 45 22 7 24 /3 = 3 DEC ■^69 5.12) P Ref:#28 Dual=#gg Elts': 27 13 22 7 3 5 24 ^ = 3 DEC -^81 5.12) P Ref:#44 Dual=#ioi Elts': 11 13 14 19 21 3 5 /? = 3 DEC ■^89 7.12) D Ref:#69 Elts': 59 29 46 67 69 /3 = 3 DEC ^101 (7.12) D Ref:#8i Elts': 123 45 86 7 24 ^ = 3 DEC -^113 (6.12) P Ref:#46 D u a l= #43 i Elts': 59 29 46 3 5 36 /? = 3 DEC ■^139 (6.12) P Ref:#49 D u a l= #63 i Elts': 27 29 14 35 373 ^ = 3 DEC -^140 (6.12) P Ref:#49 D u a l= #65 i Elts': 27 29 14 35 376 ^ = 3 DEC ^142 (6.12) P Ref:#49 D u a l= #56 i Elts': 27 29 14 35 37 11 ^ = 3 DEC ■^143 (6.12) P Ref:#49 D u a l= # n3 Elts': 27 29 14 35 37 19 /3 = 3 DEC -^144 (6.12) P Ref:#49 D u a l= #73 i Elts': 27 29 14 35 37 22 /3 = 3 DEC -^156 (6.12) P Ref:#62 D u a l= #42 i Elts': 59 13 30 3 37 20 /? = 3 DEC ■^163 (6.12) P Ref:#63 D u a l= #3 ig Elts': 59 45 30 3 37 20 = 3 DEC ■^165 (6.12) P Ref:#53 D u a l= #4 io Elts': 59 45 30 3 37 36 ^3 = 3 DEC -^173 (6.12) P Ref:#64 D u a l= #44 i Elts': 59 13 62 3 37 20 3 = 3 DEC T able A .4. Non-Iocalizable 3-structure distribution (for matroids of branch-width 3) Non-loc. 4\'s Non-loc. A's (MO) 1 (in 1 M’s) 1 (in 1 M’s) ( 5 , n j 0 (in 0 M ’s) 5 (in 4 M ’s) (6,DJ 5 (in 4 M ’s) 0 (in 0 M ’s) (5,12) 0 (in 0 M ’s) 14 (in 7 M ’s) T M 2 14 (in 7 M’s 0 (in 0 ( 0 2 9 (in 9 APs 9 (in 9 M ’s Totals 29 (in 21 M ’s) 29 (in 21 M ’s) 26 3 List A .6. Matroids that have a non-localizable 3-Structure (37) N iq: (5.10) P RefiiVy Exts=l Dual^Ajg Elts': 27 13 22 7 9 /? = 3 ACT ■^30 (5.11) P Ref:Ai7 Exts=l Duai^A^g Elts': 27 13 22 7 3 14 /? = 3 DEC -^31 (5.11) P Ref:Ai 7 Dual^Aztg Elts': 27 13 22 7 3 24 /? = 3 DEC ^34 (5.11) P Ref:#is Exts=l Dual=iV5 2 Elts': 27 13 22 7 5 17 /? - 3 DEC ^39 (5.11) P Ref:Ai9 Exts=l Dual= # 5 7 Elts': 27 13 22 7 9 11 /? = 3 DEC -/\^48 (6.11) D RehAso Exts = 6 Elts': 27 61 46 35 5 /3 = 3 DEC ^49 (6.11) D RehTVgi E x ts = 6 Elts': 27 29 14 35 37 = 3 DEC ■^52 (6.11) D Ref: # 3 4 Exts=7 Elts': 59 13 30 3 37 ^9 = 3 DEC -^57 (6.11) D Ref: # 3 9 Exts=l Elts': 59 45 14 51 5 ^ = 3 DEC -^66 (5.12) P Ref:A 2 8 D ual= N g 6 Elts': 27 13 22 7 3 5 14 - 3 DEC -^68 (5.12) P Ref:A28 Dual-Agg Elts': 27 13 22 7 3 5 21 /? = 3 DEC ^69 (5.12) P Ref: # 2 8 D u a l= A8 9 Elts': 27 13 22 7 3 5 24 /? = 3 DEC A^72 (5.12) P Ref:#29 Duai= # 9 2 Elts': 27 13 22 7 3 11 14 /? = 3 DEC A^73 (5.12) P Ref:A 3 o D u al= A9 3 Elts': 27 13 22 7 3 14 29 /? = 3 DEC iV75 (5.12) P Ref:A 3 4 D u al= N9 s Elts': 27 13 22 7 5 17 19 /? = 3 DEC A^79 (5.12) P Ref:A 3 9 D u al= A9 9 Elts': 27 13 22 7 9 11 15 /? = 3 DEC ^86 (7.12) D Ref:iV6 6 Elts': 59 93 110 67 5 /3 = 3 DEC ^ 88 (7.12) D Ref: # 6 8 Elts': 123 29 110 3 69 /î = 3 DEC -/\^89 (7.12) D Ref: # 6 9 Elts': 59 29 46 67 69 ^ - 3 DEC Nq2 (7.12) D Ref:A 7 2 Elts': 59 125 78 99 5 /? = 3 DEC ^93 (7.12) D Ref:iV7 3 Elts': 91 61 110 99 69 /? = 3 DEC -^95 (7.12) D Ref: A 7 5 E lts': 123 77 30 3 101 ^ = 3 D EC ^99 (7.12) D Ref:N 7 9 Elts': 123 109 78 115 5 /? = 3 DEC •^106 (6.12) P Ref:A 4 6 Dual=N]06 Elts': 59 29 46 3 5 13 ^ = 3 DEC -^107 (6.12) P Ref:iV4 6 Bnal= Nm Elts': 59 29 46 3 5 14 /? = 3 DEC ^113 (6.12) P Ref:iV4 6 D iial= 7Vi4 3 Elts': 59 29 46 3 5 36 yg = 3 DEC -^116 (6.12) P Ref:iV4 6 D u a l= A i3 7 Elts': 59 29 46 3 5 39 /? = 3 DEC ^126 (6.12) P Ref: # 4 7 D ual=iV io7 Elts': 59 61 14 35 5 24 /? = 3 DEC ■^129 (6.12) P Ref:A 4 7 Bual= Nm Elts': 59 61 14 35 5 38 /? = 3 DEC ^135 (6.12) P Ref: # 4 8 D ual=iV i2 9 Elts': 27 61 46 35 5 19 /I = 3 DEC ■^136 (6.12) P Ref:A 4 8 D ual=iV i3 6 Elts': 27 61 46 35 5 38 = 3 DEC #137: (6,12) P Ref: # 4 8 D u a l= # n6 Elts': 27 61 46 35 5 53 /? = 3 DEC 2 6 4 N uq (6.12) P Ref: # 4 9 D u a l= #6 i 5 Elts': 27 29 14 35 37 6 /3 3 DEC -^143 (6.12) F Ref: # 4 9 D u a l= # ii3 Elts': 27 29 14 35 37 19 /? = 3 DEC ^148 (6.12) P Ref:#5 i D u a i= #4 ig Elts': 27 45 62 3 5 11 /? - 3 DEC -^159 (6.12) P Ref: # 5 2 D ual==#i5 9 Elts': 59 13 30 3 37 29 = 3 ACT -^165 (6.12) P Ref: # 5 3 D ual=iV i4 o Elts': 59 45 30 3 37 36 = 3 DEC T able A .5. Generator distribution (p .l^ l) Generators (3,7 1 (4,7) 1 (4,81 1 (5,10) 1 (6,12) 2 Total 6 List A .7. Active generators (6) Ni: (3, 7) P Dual=#2 Elts':3 5 6 7 /3 = 3 ACT #2: (4, 7) D Ref:#i Exts=2 Elts': 11 13 14 /3 = 3 ACT #3 : (4, 8) P Ref:#2 Exts=2 Dual= # 3 Elts': 11 13 14 3 /3 = 3 ACT # 1 9 : (5,10) P Ref: # 7 Exts=l Dual=#i9 Elts': 27 13 22 7 9 ^ = 3 ACT # 1 5 8 : (6,12) P Ref: # 5 2 Dual=#i5g Elts': 59 13 30 3 37 24 /? = 3 ACT # 1 5 9 : (6,12) P Ref: # 5 2 D u a l= # is9 Elts': 59 13 30 3 37 29 ,0 = 3 ACT 266 T a b le A . 6 . Decomposition-type distribution (p ,l^ l) 1^4 # 3 # 4 F^ (4,8) 0 0 0 1 (4,9) 0 0 0 2 (5,9) 0 0 0 2 (4 , 1 0 ) 2 0 0 0 ( 6 , 1 0 ) 2 0 0 0 (4,11) 1 0 0 0 7,11) 1 0 0 0 (5 , 1 0 ) 3 1 1 4 (5,11) 16 1 0 1 __(6,11) 16 1 0 1 k l 2 ) 19 0 0 1 (7,12 19 0 0 1 (6,12) 78 2 0 3 Totals 157 5 1 16 L ist A .8. Matroids of decomposition-type (five) Nig: (5,10) P Ref:A? Exts=6 Duai=Aig Elts': 27 13 22 7 5 /3 = 3 DEC A 4 3 : (5,11) P Ref:A25 Dual^Agi Elts': 27 29 22 7 6 14 /I = 3 DEC ^ 6 1 : (6,11) D Ref:iV4 3 Exts=2 Bits': 11 61 62 35 7 /9 = 3 DEC N isq: (6,12) P RefAg] Dual^Aigg Elts': 11 61 62 35 7 3 /? = 3 DEC Nigr- (6,12) P Ref-.iVei Dual^iVJgy B its': 11 61 62 35 7 19 /? = 3 DEC L ist A .9. M atroid of decomposition-type N4 A 2 5 : (5,10) P Ref:Ag Exts=l Dual=Af25 Bits': 27 29 22 7 6 /? = 3 DEC L is t A .10. M atroids of decomposition-type Fj (six te e n ) N4 : (4, 8) P Ref:A 2 Dual=A| Elts': 11 13 14 7 /3 = 3 DEC 7V5 : (4, 9) P RefiNg Exts=3 Dual=A? Bits': 11 13 14 3 5 /3 = 3 DEC Ng: (4, 9) P Ref:A 3 Dual=A g Bits': 11 13 14 3 7 /? := 3 DEC Nr. (5, 9) D Ref-.iVs Exts=8 Bits': 27 13 22 7 /I = 3 DEC Ng: (5, 9) D Ref:Ae Exts=2 Bits': 27 29 22 7 ^ = 3 DEC #21: (5,10) P Ref: # 7 Exts=l Dual=#21 Bits': 27 13 22 7 14 ^ = 3 DEC #22: (5,10) P Ref: # 7 D ual= # 2 7 Bits': 27 13 22 7 24 ^ - 3 DEC 2 6 6 N oq: (5,10) P Ref:TVs Duai=7V26 Bits': 27 29 22 7 14 /3 = 3 DEC N 2T. (5,10) D Ref:#22 Exts=2 Bits': 11 13 14 19 21 /3 = 3 DBG #4 5 : (5,11) p Ref:#27 Bxts==l Dual= # 6 3 Bits': 11 13 14 19 21 7 /3 = 3 DBG #63: (6,11) D Ref: # 4 5 Bits': 59 45 54 7 24 ^3 = 3 DBG #83: (5,12) P Ref: # 4 5 D u a l-# io3 Bits': 11 13 14 19 21 7 22 /3 = 3 DBG #103: (7,12) D Ref:#g3 Bits': 59 109 118 7 88 = 3 DBG #124: (6,12) P Ref: # 4 7 D u a l= #2 i 4 Bits': 59 61 14 35 5 19 /3 = 3 DBG #130: (6,12) P Ref:# 4 7 D u a l= #3 io Bits': 59 61 14 35 5 48 /? = 3 DBG #178: (6,12) P Ref:# 5 5 D u a l= #7 ig Bits': 59 13 62 35 37 49 /9 = 3 DBG 2 6 7 T able A .7. Statistical information about the distribution of the number of bases M ’s Min M ax M edian Table A.7. (Continued) (P ,|g|) /l /2 h (3,7 1.000 1.000 1 . 0 0 0 (4, 7) 1.000 1.000 1 . 0 0 0 (4,8 1.000 1.000 1 . 0 0 0 (4,9 1.000 1.000 1 . 0 0 0 (5,9) 1.000 1.000 1 . 0 0 0 (4,10) 0.333 1.000 1 . 0 0 0 6,10) 0.333 1.000 1 . 0 0 0 (4,11) 1.000 1.000 1 . 0 0 0 (7,11) 1.000 1.000 1 . 0 0 0 (5 ,10) 0.727 0.909 1 . 0 0 0 5,11 0.667 0.944 1 . 0 0 0 (6 , 1 1 ) 0.667 0.944 1 . 0 0 0 5 , 1 2 ) 0.750 0.950 1 . 0 0 0 (7, 12) 0.750 0.950 1 . 0 0 0 (6, 12) 0.682 0.988 0^#8 268 A histogram for the probabihty distribution of the sizes of bases of matroids, of rank 6 that consist of 12 elements, is given below. The probabihties range à à - P R 0 B A B 1 L I T Y 352 576 Size of base Figure A .2. A histogram for probabihties of base sizes. 2 6 9 T a b le A .8 . Statistical information about the distribution of the number of column pseudo-orbits (A 1^1) M’s Min Max Median u (3,7) 1 1 1 1.00 1.00 0.00 (4,7) 1 1 1 1.00 1.00 0.00 (4,8 2 1 3 2.00 2.00 1.00 4 ,9 2 2 4 3.00 3.00 1.00 (5,9) 2 2 4 3.00 3.00 1.00 (4 ,lO) 3 2 3 3.00 2.67 0.47 (6,10) 3 2 3 3.00 2.67 0.47 (4,11) 1 2 2 2.00 2.00 0.00 (7,11) 1 2 2 2.00 2.00 0.00 (5,10) 11 1 6 4.00 3.82 1.53 (5,11) 18 2 8 4.00 4.50 1.71 ( 6 ,ll) 18 2 8 4.00 4.50 1.71 (5,12) 20 1 9 4.00 4.85 1.82 (7,12) 20 1 9 4.00 4.85 1.82 (6 , 1 2 ) 85 1 10 6.00 5.74 1.99 Table A.8. (Continued) (A 1^1) h h h (3,7 1 . 0 0 0 1.000 1.000 (4,7) 1 . 0 0 0 1.000 1.000 (4,8) 1 . 0 0 0 1.000 1.000 (4, 9 ) 1 . 0 0 0 1.000 1.000 (5, 9 ) 1 . 0 0 0 1.000 1.000 (4, 10) 0.667 1.000 1.000 (6,10) 0.667 1.000 1.000 (4,11) 1.000 1.000 1.000 (7,11 1.000 1.000 1.000 (5, 10) 0.636 1.000 1.000 (5 , 1 1 ) 0.722 0.944 1.000 6, 11) 0.722 0.944 1.000 (5,12) 0.650 0.900 1.000 (7, 12) 0.650 0.900 1.000 (6,12) 0.682 0.953 1.000 270 A histogram for the probabihties of column pseudo-orbit sizes of matroids of rank 6 that consist of 12 elements, is given below. (20/85) (14/85) P R 0 (12/85) (12/85) B A B I L (9/85) I T (8/85) Y (6/85) (3/85) (1/85) 012345678 9 10 Size of column pseudo-orbit Figure A .3. A histogram for probabihties of column pseudo-orbit sizes A P P E N D I X B An Addition Table For Coded Binary Vectors Table B .l. The addition table + 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1 0 3 2 5 4 7 6 9 8 11 10 13 12 15 14 2 3 0 1 6 7 4 5 10 11 8 9 14 15 12 13 3 2 1 0 7 6 5 4 11 10 9 8 15 14 13 12 4 5 6 7 0 1 2 3 12 13 14 15 8 9 10 11 5 4 7 6 1 0 3 2 13 12 15 14 9 8 11 10 6 7 4 5 2 3 0 1 14 15 12 13 10 11 8 9 7 6 5 4 3 2 1 0 15 14 13 12 11 10 9 8 8 9 10 11 12 13 14 15 0 1 2 3 4 5 6 7 9 8 11 10 13 12 15 14 1 0 3 2 5 4 7 6 10 11 8 9 14 15 12 13 2 3 0 1 6 7 4 5 11 10 9 8 15 14 13 12 3 2 1 0 7 6 5 4 12 13 14 15 8 9 10 11 4 5 6 7 0 1 2 3 13 12 15 14 9 8 11 10 5 4 7 6 1 0 3 2 14 15 12 13 10 11 8 9 6 7 4 5 2 3 0 1 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 17 16 19 18 21 20 23 22 25 24 27 26 29 28 31 30 18 19 16 17 22 23 20 21 26 27 24 25 30 31 28 29 19 18 17 16 23 22 21 20 27 26 25 24 31 30 29 28 20 21 22 23 16 17 18 19 28 29 30 31 24 25 26 27 21 20 23 22 17 16 19 18 29 28 31 30 25 24 27 26 22 23 20 21 18 19 16 17 30 31 28 29 26 27 24 25 23 22 21 20 19 18 17 16 31 30 29 28 27 26 25 24 24 25 26 27 28 29 30 31 16 17 18 19 20 21 22 23 25 24 27 26 29 28 31 30 17 16 19 18 21 20 23 22 26 27 24 25 30 31 28 29 18 19 16 17 22 23 20 21 27 26 25 24 31 30 29 28 19 18 17 16 23 22 21 20 28 29 30 31 24 25 26 27 20 21 22 23 16 17 18 19 29 28 31 30 25 24 27 26 21 20 23 22 17 16 19 18 30 31 28 29 26 27 24 25 22 23 20 21 18 19 16 17 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 271 Table B .l. (Continued) + 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 1 17 16 19 18 21 20 23 22 25 24 27 26 29 28 31 30 2 18 19 16 17 22 23 20 21 26 27 24 25 30 31 28 29 3 19 18 17 16 23 22 21 20 27 26 25 24 31 30 29 28 4 20 21 22 23 16 17 18 19 28 29 30 31 24 25 26 27 5 21 20 23 22 17 16 19 18 29 28 31 30 25 24 27 26 6 22 23 20 21 18 19 16 17 30 31 28 29 26 27 24 25 7 23 22 21 20 19 18 17 16 31 30 29 28 27 26 25 24 8 24 25 26 27 28 29 30 31 16 17 18 19 20 21 22 23 9 25 24 27 26 29 28 31 30 17 16 19 18 21 20 23 22 10 26 27 24 25 30 31 28 29 18 19 16 17 22 23 20 21 11 27 26 25 24 31 30 29 28 19 18 17 16 23 22 21 20 12 28 29 30 31 24 25 26 27 20 21 22 23 16 17 18 19 13 29 28 31 30 25 24 27 26 21 20 23 22 17 16 19 18 14 30 31 28 29 26 27 24 25 22 23 20 21 18 19 16 17 15 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 16 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 17 1 0 3 2 5 4 7 6 9 8 11 10 13 12 15 14 18 2 3 0 1 6 7 4 5 10 11 8 9 14 15 12 13 19 3 2 1 0 7 6 5 4 11 10 9 8 15 14 13 12 20 4 5 6 7 0 1 2 3 12 13 14 15 8 9 10 11 21 5 4 7 6 1 0 3 2 13 12 15 14 9 8 11 10 22 6 7 4 5 2 3 0 1 14 15 12 13 10 11 8 9 23 7 6 5 4 3 2 1 0 15 14 13 12 11 10 9 8 24 8 9 10 11 12 13 14 15 0 1 2 3 4 5 6 7 25 9 8 11 10 13 12 15 14 1 0 3 2 5 4 7 6 26 10 11 8 9 14 15 12 13 2 3 0 1 6 7 4 5 27 11 10 9 8 15 14 13 12 3 2 1 0 7 6 5 4 28 12 13 14 15 8 9 10 11 4 5 6 7 0 1 2 3 29 13 12 15 14 9 8 11 10 5 4 7 6 1 0 3 2 30 14 15 12 13 10 11 8 9 6 7 4 5 2 3 0 1 31 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 272 Table B .l. (Continued) + 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 1 33 32 35 34 37 36 39 38 41 40 43 42 45 44 47 46 2 34 35 32 33 38 39 36 37 42 43 40 41 46 47 44 45 3 35 34 33 32 39 38 37 36 43 42 41 40 47 46 45 44 4 36 37 38 39 32 33 34 35 44 45 46 47 40 41 42 43 5 37 36 39 38 33 32 35 34 45 44 47 46 41 40 43 42 6 38 39 36 37 34 35 32 33 46 47 44 45 42 43 40 41 7 39 38 37 36 35 34 33 32 47 46 45 44 43 42 41 40 8 40 41 42 43 44 45 46 47 32 33 34 35 36 37 38 39 9 41 40 43 42 45 44 47 46 33 32 35 34 37 36 39 38 10 42 43 40 41 46 47 44 45 34 35 32 33 38 39 36 37 11 43 42 41 40 47 46 45 44 35 34 33 32 39 38 37 36 12 44 45 46 47 40 41 42 43 36 37 38 39 32 33 34 35 13 45 44 47 46 41 40 43 42 37 36 39 38 33 32 35 34 14 46 47 44 45 42 43 40 41 38 39 36 37 34 35 32 33 15 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 16 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 17 49 48 51 50 53 52 55 54 57 56 59 58 61 60 63 62 18 50 51 48 49 54 55 52 53 58 59 56 57 62 63 60 61 19 51 50 49 48 55 54 53 52 59 58 57 56 63 62 61 60 20 52 53 54 55 48 49 50 51 60 61 62 63 56 57 58 59 21 53 52 55 54 49 48 51 50 61 60 63 62 57 56 59 58 22 54 55 52 53 50 51 48 49 62 63 60 61 58 59 56 57 23 55 54 53 52 51 50 49 48 63 62 61 60 59 58 57 56 24 56 57 58 59 60 61 62 63 48 49 50 51 52 53 54 55 25 57 56 59 58 61 60 63 62 49 48 51 50 53 52 55 54 26 58 59 56 57 62 63 60 61 50 51 48 49 54 55 52 53 27 59 58 57 56 63 62 61 60 51 50 49 48 55 54 53 52 28 60 61 62 63 56 57 58 59 52 53 54 55 48 49 50 51 29 61 60 63 62 57 56 59 58 53 52 55 54 49 48 51 50 30 62 63 60 61 58 59 56 57 54 55 52 53 50 51 48 49 31 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 273 Table B .l. (Continued) + 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 1 49 48 51 50 53 52 55 54 57 56 59 58 61 60 63 62 2 50 51 48 49 54 55 52 53 58 59 56 57 62 63 60 61 3 51 50 49 48 55 54 53 52 59 58 57 56 63 62 61 60 4 52 53 54 55 48 49 50 51 60 61 62 63 56 57 58 59 5 53 52 55 54 49 48 51 50 61 60 63 62 57 56 59 58 6 54 55 52 53 50 51 48 49 62 63 60 61 58 59 56 57 7 55 54 53 52 51 50 49 48 63 62 61 60 59 58 57 56 8 56 57 58 59 60 61 62 63 48 49 50 51 52 53 54 55 9 57 56 59 58 61 60 63 62 49 48 51 50 53 52 55 54 10 58 59 56 57 62 63 60 61 50 51 48 49 54 55 52 53 11 59 58 57 56 63 62 61 60 51 50 49 48 55 54 53 52 12 60 61 62 63 56 57 58 59 52 53 54 55 48 49 50 51 13 61 60 63 62 57 56 59 58 53 52 55 54 49 48 51 50 14 62 63 60 61 58 59 56 57 54 55 52 53 50 51 48 49 15 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 16 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 17 33 32 35 34 37 36 39 38 41 40 43 42 45 44 47 46 18 34 35 32 33 38 39 36 37 42 43 40 41 46 47 44 45 19 35 34 33 32 39 38 37 36 43 42 41 40 47 46 45 44 20 36 37 38 39 32 33 34 35 44 45 46 47 40 41 42 43 21 37 36 39 38 33 32 35 34 45 44 47 46 41 40 43 42 22 38 39 36 37 34 35 32 33 46 47 44 45 42 43 40 41 23 39 38 37 36 35 34 33 32 47 46 45 44 43 42 41 40 24 40 41 42 43 44 45 46 47 32 33 34 35 36 37 38 39 25 41 40 43 42 45 44 47 46 33 32 35 34 37 36 39 38 26 42 43 40 41 46 47 44 45 34 35 32 33 38 39 36 37 27 43 42 41 40 47 46 45 44 35 34 33 32 39 38 37 36 28 44 45 46 47 40 41 42 43 36 37 38 39 32 33 34 35 29 45 44 47 46 41 40 43 42 37 36 39 38 33 32 35 34 30 46 47 44 45 42 43 40 41 38 39 36 37 34 35 32 33 31 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 274 Table B .l. (Continued) + 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 33 32 35 34 37 36 39 38 41 40 43 42 45 44 47 46 34 35 32 33 38 39 36 37 42 43 40 41 46 47 44 45 35 34 33 32 39 38 37 36 43 42 41 40 47 46 45 44 36 37 38 39 32 33 34 35 44 45 46 47 40 41 42 43 37 36 39 38 33 32 35 34 45 44 47 46 41 40 43 42 38 39 36 37 34 35 32 33 46 47 44 45 42 43 40 41 39 38 37 36 35 34 33 32 47 46 45 44 43 42 41 40 40 41 42 43 44 45 46 47 32 33 34 35 36 37 38 39 41 40 43 42 45 44 47 46 33 32 35 34 37 36 39 38 42 43 40 41 46 47 44 45 34 35 32 33 38 39 36 37 43 42 41 40 47 46 45 44 35 34 33 32 39 38 37 36 44 45 46 47 40 41 42 43 36 37 38 39 32 33 34 35 45 44 47 46 41 40 43 42 37 36 39 38 33 32 35 34 46 47 44 45 42 43 40 41 38 39 36 37 34 35 32 33 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 49 48 51 50 53 52 55 54 57 56 59 58 61 60 63 62 50 51 48 49 54 55 52 53 58 59 56 57 62 63 60 61 51 50 49 48 55 54 53 52 59 58 57 56 63 62 61 60 52 53 54 55 48 49 50 51 60 61 62 63 56 57 58 59 53 52 55 54 49 48 51 50 61 60 63 62 57 56 59 58 54 55 52 53 50 51 48 49 62 63 60 61 58 59 56 57 55 54 53 52 51 50 49 48 63 62 61 60 59 58 57 56 56 57 58 59 60 61 62 63 48 49 50 51 52 53 54 55 57 56 59 58 61 60 63 62 49 48 51 50 53 52 55 54 58 59 56 57 62 63 60 61 50 51 48 49 54 55 52 53 59 58 57 56 63 62 61 60 51 50 49 48 55 54 53 52 60 61 62 63 56 57 58 59 52 53 54 55 48 49 50 51 61 60 63 62 57 56 59 58 53 52 55 54 49 48 51 50 62 63 60 61 58 59 56 57 54 55 52 53 50 51 48 49 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 27 5 Table B .l. (Continued) + 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 33 49 48 51 50 53 52 55 54 57 56 59 58 61 60 63 62 34 50 51 48 49 54 55 52 53 58 59 56 57 62 63 60 61 35 51 50 49 48 55 54 53 52 59 58 57 56 63 62 61 60 36 52 53 54 55 48 49 50 51 60 61 62 63 56 57 58 59 37 53 52 55 54 49 48 51 50 61 60 63 62 57 56 59 58 38 54 55 52 53 50 51 48 49 62 63 60 61 58 59 56 57 39 55 54 53 52 51 50 49 48 63 62 61 60 59 58 57 56 40 56 57 58 59 60 61 62 63 48 49 50 51 52 53 54 55 41 57 56 59 58 61 60 63 62 49 48 51 50 53 52 55 54 42 58 59 56 57 62 63 60 61 50 51 48 49 54 55 52 53 43 59 58 57 56 63 62 61 60 51 50 49 48 55 54 53 52 44 60 61 62 63 56 57 58 59 52 53 54 55 48 49 50 51 45 61 60 63 62 57 56 59 58 53 52 55 54 49 48 51 50 46 62 63 60 61 58 59 56 57 54 55 52 53 50 51 48 49 47 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 48 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 49 33 32 35 34 37 36 39 38 41 40 43 42 45 44 47 46 50 34 35 32 33 38 39 36 37 42 43 40 41 46 47 44 45 51 35 34 33 32 39 38 37 36 43 42 41 40 47 46 45 44 52 36 37 38 39 32 33 34 35 44 45 46 47 40 41 42 43 53 37 36 39 38 33 32 35 34 45 44 47 46 41 40 43 42 54 38 39 36 37 34 35 32 33 46 47 44 45 42 43 40 41 55 39 38 37 36 35 34 33 32 47 46 45 44 43 42 41 40 56 40 41 42 43 44 45 46 47 32 33 34 35 36 37 38 39 57 41 40 43 42 45 44 47 46 33 32 35 34 37 36 39 38 58 42 43 40 41 46 47 44 45 34 35 32 33 38 39 36 37 59 43 42 41 40 47 46 45 44 35 34 33 32 39 38 37 36 60 44 45 46 47 40 41 42 43 36 37 38 39 32 33 34 35 61 45 44 47 46 41 40 43 42 37 36 39 38 33 32 35 34 62 46 47 44 45 42 43 40 41 38 39 36 37 34 35 32 33 63 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 276 Table B .l. (Continued) + 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 32 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 33 1 0 3 2 5 4 7 6 9 8 11 10 13 12 15 14 34 2 3 0 1 6 7 4 5 10 11 8 9 14 15 12 13 35 3 2 1 0 7 6 5 4 11 10 9 8 15 14 13 12 36 4 5 6 7 0 1 2 3 12 13 14 15 8 9 10 11 37 5 4 7 6 1 0 3 2 13 12 15 14 9 8 11 10 38 6 7 4 5 2 3 0 1 14 15 12 13 10 11 8 9 39 7 6 5 4 3 2 1 0 15 14 13 12 11 10 9 8 40 8 9 10 11 12 13 14 15 0 1 2 3 4 5 6 7 41 9 8 11 10 13 12 15 14 1 0 3 2 5 4 7 6 42 10 11 8 9 14 15 12 13 2 3 0 1 6 7 4 5 43 11 10 9 8 15 14 13 12 3 2 1 0 7 6 5 4 44 12 13 14 15 8 9 10 11 4 5 6 7 0 1 2 3 45 13 12 15 14 9 8 11 10 5 4 7 6 1 0 3 2 46 14 15 12 13 10 11 8 9 6 7 4 5 2 3 0 1 47 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 48 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 49 17 16 19 18 21 20 23 22 25 24 27 26 29 28 31 30 50 18 19 16 17 22 23 20 21 26 27 24 25 30 31 28 29 51 19 18 17 16 23 22 21 20 27 26 25 24 31 30 29 28 52 20 21 22 23 16 17 18 19 28 29 30 31 24 25 26 27 53 21 20 23 22 17 16 19 18 29 28 31 30 25 24 27 26 54 22 23 20 21 18 19 16 17 30 31 28 29 26 27 24 25 55 23 22 21 20 19 18 17 16 31 30 29 28 27 26 25 24 56 24 25 26 27 28 29 30 31 16 17 18 19 20 21 22 23 57 25 24 27 26 29 28 31 30 17 16 19 18 21 20 23 22 58 26 27 24 25 30 31 28 29 18 19 16 17 22 23 20 21 59 27 26 25 24 31 30 29 28 19 18 17 16 23 22 21 20 60 28 29 30 31 24 25 26 27 20 21 22 23 16 17 18 19 61 29 28 31 30 25 24 27 26 21 20 23 22 17 16 19 18 62 30 31 28 29 26 27 24 25 22 23 20 21 18 19 16 17 63 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 277 Table B .l. (Continued) + 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 32 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 33 17 16 19 18 21 20 23 22 25 24 27 26 29 28 31 30 34 18 19 16 17 22 23 20 21 26 27 24 25 30 31 28 29 35 19 18 17 16 23 22 21 20 27 26 25 24 31 30 29 28 36 20 21 22 23 16 17 18 19 28 29 30 31 24 25 26 27 37 21 20 23 22 17 16 19 18 29 28 31 30 25 24 27 26 38 22 23 20 21 18 19 16 17 30 31 28 29 26 27 24 25 39 23 22 21 20 19 18 17 16 31 30 29 28 27 26 25 24 40 24 25 26 27 28 29 30 31 16 17 18 19 20 21 22 23 41 25 24 27 26 29 28 31 30 17 16 19 18 21 20 23 22 42 26 27 24 25 30 31 28 29 18 19 16 17 22 23 20 21 43 27 26 25 24 31 30 29 28 19 18 17 16 23 22 21 20 44 28 29 30 31 24 25 26 27 20 21 22 23 16 17 18 19 45 29 28 31 30 25 24 27 26 21 20 23 22 17 16 19 18 46 30 31 28 29 26 27 24 25 22 23 20 21 18 19 16 17 47 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 48 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 49 1 0 3 2 5 4 7 6 9 8 11 10 13 12 15 14 50 2 3 0 1 6 7 4 5 10 11 8 9 14 15 12 13 51 3 2 1 0 7 6 5 4 11 10 9 8 15 14 13 12 52 4 5 6 7 0 1 2 3 12 13 14 15 8 9 10 11 53 5 4 7 6 1 0 3 2 13 12 15 14 9 8 11 10 54 6 7 4 5 2 3 0 1 14 15 12 13 10 11 8 9 55 7 6 5 4 3 2 1 0 15 14 13 12 11 10 9 8 56 8 9 10 11 12 13 14 15 0 1 2 3 4 5 6 7 57 9 8 11 10 13 12 15 14 1 0 3 2 5 4 7 6 58 10 11 8 9 14 15 12 13 2 3 0 1 6 7 4 5 59 11 10 9 8 15 14 13 12 3 2 1 0 7 6 5 4 60 12 13 14 15 8 9 10 11 4 5 6 7 0 1 2 3 61 13 12 15 14 9 8 11 10 5 4 7 6 1 0 3 2 62 14 15 12 13 10 11 8 9 6 7 4 5 2 3 0 1 63 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 278 A P P E N D I X C Outline of the Computer Program The purpose of this appendix is to give a rough outhne of the computer pro gram, the output of which was presented in Appendix A. The entire program was written in the programming language Pascal, and was compiled and run on the IBM 3090 which was provided by Academic Computing Services (ACS) at The Oliio State University. The editor used was WYLBUR, also provided by ACS. Earher versions of this program utihzed the above-mentioned resources at ACS, and were also compiled and run on the CRAY at the Ohio Supercomputer Center (CSC). The input to the program consists of 2 integers and the vectorial representa tions of the polygon matroids of 5 graphs. The values of the input integers are taken by the variables m ostelts and docbit, respectively. The polygon matroids are those of the octacube, TTg, augmented Ag 3 graph, the dual of the 3-prism and the augmented 3-prism, respectively. The variable m ostelts gives the up per bound for the number of elements of any matroid that is generated by the program. For the output given in Appendix A, mostelts = 12. The variable docbit merely controls the printing of documentation that appears before the hsts and tables of the output. The actual computer program contains about 100 subroutines, and is quite comphcated. Due to this, we only describe the process by which List A1.2 was obtained, and the more important subroutines, in a simpUfied version of the pro gram. This description is in Enghsh, and contains no Pascal code or pseudo code. Accordingly, when we discuss variables and subroutines we shall not con cern ourselves with the rules of Pascal. During the course of this description we 279 2 8 0 treat certain groups of data structures and subroutines as a single data structure and subroutine, respectively, in order to present the essence of the program in a simple manner. We refer to any such data structure as a compound array. Let m g en era te denote the subroutine of the program which generates and saves the binary matroids of List A1.2. From Appendix A, we recall that List A1.2 contains an isomorphic copy of every non-regular 3-connected binary ma troid that has at most 1 2 elements, and either belongs to ^ 3 or is an obstacle to that ideal. The subroutine m g en era te generates these binary matroids by be ginning with the Fano matroid F%, and performing the 2 operations of duahzation and single element extension, respectively. To facihtate this description, we now introduce the notion of a block. Let r and 5 be positive integers such that r < S. Then the set of all non-regular 3- connected binary matroids of rank r that consist of S elements, is called the block of shape (r, 5), and is denoted by Blk(r, 5). The integers r and 5 are called the rank and the size of BIk(r, S). The subroutine m g en era te begins with Blk(3, 7), wliich consist of F-j. Thereafter, it generates non-regular matroids one block at a time. We first describe how m g en era te proceeds from one matroid to another, and then explain the manner in which it moves from one block to another. In order to process any matroid M, m g en era te must first obtain a vectorial presentation of M. In the case of Fj, which is the first matroid of List A1.2, m g en era te calls the subroutine m g etF a n o from which it obtains the vectorial presentation of F 7 . Thereafter it obtains vectorial presentations of subsequently generated matroids by calling the subroutines that perform the 2 operations of duahzation and single element extension, respectively. Because m o ste lts —12, every matroid generated by m g en era te has at most 12 elements. Suppose a m atroid M is generated by mgenerate. Then a vectorial presentation of M is stored in the compound array mpresentation. The matroid M is accepted into List A1.2 if and only if M is not isomorphic to any of the matroids that are already in List A1.2, and either M E . ^ 3 or M is an obstacle to that ideal. If M is generated as a primal then in order to determine whether M satisfies these criteria, m g en era te first finds all the information about M that appears in the output. If M is generated as a dual then the corresponding information is obtained from the parent of M. The above-mentioned information about M 281 is stored in the compound array m sta ts. The branch-width of M is determined by the subroutine mbranch-widthvalue and stored in the array m branch- w id th . After finding the information that is stored in mstats, mgenerate calls the subroutine misocheck, which determines whether M is isomorphic to any matroid that is already in List A1.2. If M is generated after i matroids have been accepted into List Al.2 where i > 0, then information about M is stored in the (z + 1)*'^ memory location of m sta ts. The subroutine m g en era te implements the acceptance or rejection of any matroid generated by it, by means of the integer variable m n e x t, as follows. Suppose at a certain stage k matroids have been accepted into List Al.2. Then m n e x t is set equal to A; + 1. (So at the very beginning mnext = 1). Let M be the next matroid that is generated by mgenerate. Then as explained in the previous paragraph, all the information about M is stored in the mnext*'* = {k + I)*** memory location of each of those respective compound arrays. If M is accepted into List Al.2 then m g en era te increments the value of m n ex t by 1, to let m n ex t = k+2. If on the other hand, M is rejected then m n ext is not incremented, and m g en er a te either generates another matroid or stops the generation of matroids. Thus if M is accepted then the information about M is saved at the {k + 1)*'* memory locations of the aforementioned respective arrays, but if M is rejected then the information about M is automatically discarded because it wiU be overwritten by the corresponding information for the next matroid generated. List Al.2 is complete when m g en era te stops generating matroids, and then List Al.2 consists of m n ex t -1 m atroids. We next explain how m g en era te generates the different blocks, and specify the sequence in which this generation occurs. Apart from T? wliich is gener ated by caUing the subroutine mgetFano, mgenerate generates every other m atroid M from its jiarent P{M) say, either by performing the duahzation or a single element extension of the latter. The matroid P[M) has to be a ma troid that is in List Al.2. Also the dual of P[M) is constructed and processed by m g en era te before any of the single element extensions of P{M) are con structed. While m g en era te obtains only 1 matroid by dualizing P{M), it m ust obtain — 1 single element extensions of P{M). Thus, in general there are rank-wise exponentially many single element extensions to be constructed. The 282 exception to this occurs at the very beginning with when no single element extensions can be performed. With one notable exception, M and P{M) belong to different blocks. The exception occurs when p{P{M)) — p*{P{M)) and M is obtained by dualizing P{M). In that case both M and P{M) belong to the same block, Blk(|5(M )|/2, |5(M)|). When generating matroids, m g en era te considers all the matroids of a partic ular blockB that have branch-width 3 and have been selected by List Al.2, as parents. The matroids of this selected set X are considered in the order in wliich they appear in List Al.2. Suppose B = Blk(r, 5). If r = 5/2 then when the matroids of X are dualized and processed by mgenerate, the set X may be ex panded. Let X' be the expanded set if r = n/2, and let X ' = X if r ^ 5 /2 . The subroutine m g en er a te will consider the matroids of another block as parents, only after it has completed processing, first the duahzations, and then the single element extensions, of every matroid in Suppose that matroids in the blocks B* and are generated when m g en era te processes the duahzations and the single element extensions, respectively, of the matroids of B that are selected by List Al.2. Then B* — BIk(5 — r, 5) and = Blk(r, 5 -f 1). If r ^ 5 /2 then every matroid of B* that is generated as the dual of a selected matroid of B is selected into List Al.2. In these instances m g en era te simply enters those dual matroids into List Al.2. On the other hand, if r = 5/2 then B* — B. Therefore, m g en era te m ust use m isoch eck and check if a matroid that is obtained by duahzing is non-isomorpliic to every matroid of B that has already been selected by List Al.2, before entering that dual matroid into List Al.2. The shapes of the blocks, in the sequence in which these blocks are generated by m g en era te are (3,7), (4,7), (4,8), (4,9), (5,9), (4,10), (6,10), (4,11), (7,11), (4,12), (8,12), (5,10), (5,11), (6,11), (5,12), (7,12) and (6,12). Figure C.l illustrates how matroids of different blocks are generated from matroids of other blocks. An arrow from (a, 6) to (c, d) signifies that matroids of Blk(c, d) are generated from those of Blk(a, 5). The generation was by duahzation or single element extension if the arrow is labelled by an asterisk or the character string ‘S.E.E’, respectively. There are no matroids of Blk(4,12) in List Al.2. This is because every matroid belonged to this block that was generated, was a non obstacle of branch-width 4. 283 (8,12)< ------(4,12) (7,12)<------(5,12) , 12) A A S.E.E. S.E.E. S.E.E. (4, 11) (5,11) ^ ( 6, 11) A A S.E.E. S.E.E. (6,10)<------(4,10) + 4 (5,10) A S.E.E. S.E.E. (4, 9) (5,9) A S.E.E. * A (4^8) S.E.E. (3, 7) (4,7) Figure C .l. The generation of 3-connected binary matroids by the program The subroutine m branch-w idthvalue is a Pascal function which finds /3(M), where M is a matroid that is generated by m generate. Since Fj is a minor of M it follows that /3{M) > 3. Also since all matroids of branch-width 4 that are not obstacles to ^ 3 are discarded, j3{M) < 4. Due to this m b ra n c h -w id th v a lu e is implemented by examining whether M is completely 3-separable or not. Using the central-edge-lemma, every separate search for the complete 3-separability of M begins by considering a separation {A,B) of M such that |A| > |S(M )|/3 < \B\. Thereafter, A and B are tested for complete 3-separabihty, as required by the definition of complete-separability. If M is not completely 3-separable then then the single-element deletion and contraction minors of M are tested for branch-width, again using m hranch-w idthvalue, in order to determine whether M is an obstacle to ^ 3 . The subroutine mbranch-widthvalue is also used by the subroutine mlo- calizability, which determines whether a triad or a triangle of a matroid M of 2 8 4 branch-width 3, is locahzable. If X is a triangle of M then X is a triad of M*, and X is locahzable for M if and only if X is locahzable for M*. Therefore, since it is sufficient for mlocalizability to be able to determine the locahzabhty of a triad, suppose X is a triad of M, and let X — {a,6,c}. The subroutine m lo calizability constructs the matroid M-f, which is the single element extension of M by the element corresponding to the vector a + b. T hen X U {a -f- 6} is a wye-delta of M-f.. Since a + b belongs to the span of X, it foUows that X is non-locahzable if > 3. Conversely, if we assume that every wye-delta of M-f is locahzable then X is locahzable if P{M) — 3. The essential process in the subroutine m isoch eck verifies whether 2 matroids M l and M2 that are generated by mgenerate, are isomorphic to each other or not. This process begins with the comparison of the data stored in the com pound array m sta ts, at the locations and Z2 which correspond to M i and M2, respectively. In this comparison only the cardinahties of the various structures and sets are compared. If the values obtained for Mi and Mo are different for one of these cardinahties, then Mi % M 2 . So, suppose otherwise. In that case m isoch eck examines the row spaces of Mi and M 2 , because Mi = M2 if and only if one of these rowspaces can be transformed into the other by performing row and column permutations. Due to time constraints it is not possible to ex amine whether such a transformation exists, without first rearranging the each rowspace into a standard form. A standard form for a rowspace is obtained as foUows. Suppose M is a matroid generated by m g en era te and V is the rowspace of M . A standard form for V is obtained by performing 2 permutations of the rows of V and one permutation of the columns of V. The first is a permutation of the rows. This is followed by a permutation of the columns wliich depends on the first permutation, and the last permutation is a rearrangement of the rows which depends on the first 2 permutations. The first permutation rearranges the rows of V so that the rows whose supports have the same cardinality appear in contiguous blocks. In this rearrangement the rows whose supports have smaller cardinality occur first. It can be proved that the support of every column of V has cardinality The permutation of the columns of V is determined by the blocks, into which the rows of V were partitioned by the first permutation. 2 8 5 Let the sets of coordinates of these blocks of rows he R\ ... Rt where t > 1 , and let C be a column of V. Also let cv{C) = (|i?i fl Supp(C')|... fl Supp(C')|). Then cv[C) is called the column support vector of C. The column permutation first rearranges the columns of V so that columns that have the same column support vector appear in a contiguous block. The order in which these blocks are to be rearranged is determined as follows. Throughout the program execution, as column support vectors arise they are stored in an array mcolsuppvectors, with each such vector stored exactly once in this array. A linear ordering is then estabhshed among the column support vectors, with those that appear earlier in the array mcolsuppvectors being defined to be less than those that appear later. Since every column in any block has the same column support vector, and columns in different blocks have different column support vectors it is now possible to rearrange the blocks of columns, with the blocks with smaller column support vectors preceding those with larger column support vectors. The final permutation of rows is effected in a manner analogous to that in which the column permutation was realized. The permutation of the rows of V is determined by the blocks into which the columns of V were partitioned by the second permutation. For every row R oi V a. row support vector is constructed in a manner analogous to the construction of column support vectors. Thereafter these row support vectors are ordered and the rows are rearranged in contiguous blocks, with the blocks with smaller row support vectors appearing before those with larger row support vectors. The blocks of rows and columns of this standard form of V are called the pseudo row orbits and pseudo column orbits, respectively of V, because they are unions of row orbits and columns orbits, respectively of V. Let Vi be a standard form of the row space of M{ for 1 < î < 2, obtained as explained in the previous paragraph. The subroutine m isoch eck proceeds to investigate if Vi can be transformed into Vb hy performing row and column permutations, as follows. It first checks the row support vectors and column support vectors of V] and If a transformation is possible then the rows and columns of Vi and V2 must have the same row support vectors and column support vectors, respectively, in the same respective multiphcities. Suppose Vi and V2 satisfy this condition. Then m isoch eck keeps V\ fixed, and rearranges the columns and rows of Lg, in order to obtain Vj, keeping every column and row 28 6 within its respective pseudo orbit. This rearrangement of V2 begins by matching a column of V2 with the first column of Vi, and then increasing the number of matched columns, one at a time. If misocheck fails to match columns after succeeding up to some point then it backtracks and proceeds. We conclude this description with a brief discussion of some routines that may be viewed as peripheral. These are the routines that are concerned with input and output (I/O), data gathering for tables and hsts, and initialization. Apart from reading input and writing output, which are the basic I/O routines, these routines also include those which prepare data for output. The routine that converts decimal digits into hexadecimal digits is an example of such. Routines that gather data for hsts and tables do so by searching arrays for matroids that have certain properties or structures. The obvious purpose of initiahzation routines is to initiahze variables. Apart from initializing integer variables, these routines also initiahze some arrays. We give three examples of such instances. The first two examples involve binary vectors. With a few exceptions, the program deals with binary vectors as integers, that are determined by the encoding process explained in Section 1.6. The integer corresponding to a given binary vector is determined by a procedure. While that procedure is simple, a procedure to find the binary vector that corresponds to a given positive integer would be somewhat comphcated. So, instead of using such a procedure, the program initiahzes a 2-dimensional array with 2^“ = 1024 columns so that the column consist of the binary vector that corresponds to the integer i — 1. Using this array, it is now possible to read off the binary vector that corresponds to any positive integer i, where 0 < i < 1023. The program also initiahzes an addition table giving all vector sums ‘from’ 0 + 0 ‘to ’ 255 + 255. This table makes it possible to do calculations with binary vectors, while they are represented as integers, and obtain the resulting binary vectors also as integers. The third example is useful in finding the branch-width of a binary matroid M. In constructing a branch decomposition of M, it is often necessary to consider all the ways in which a certain subset of M may be divided into 2 subsets. To help in this process, arrays are initiahzed to store all possible bifurcations of sets of relevant sizes. Due to the many computations in this program, it is important to save execution time, as much as possible. By initiahzing these arrays the 2 8 7 program saves execution time which is costly, at the expense of memory which is relatively inexpensive. 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