DOCSLIB.ORG

Binary Matroids and Fundamental Graphs

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Binary Matroids and Fundamental Graphs Binary matroids and Fundamental graphs O-joung Kwon May 3, 2012 Binary matroids and Fundamental graphs O-joung Kwon May 3, 2012 Matroid E : set, I : a set of subsets of E. A pair M = (E; I) is a matroid on E if it satisfies the following three axioms. (T1) I= 6 ;. (T2) If X 2 I and Y ⊆ X, then Y 2 I. (T3) Let X; Y 2 I and jXj = jY j + 1. Then there exists x 2 X n Y such that Y [ fxg 2 I. X 2 I : an independent set of M. An independent set X of M is a base of M if it is a maximal independent set. Matroid E : set, I : a set of subsets of E. A pair M = (E; I) is a matroid on E if it satisfies the following three axioms. (T1) I= 6 ;. (T2) If X 2 I and Y ⊆ X, then Y 2 I. (T3) Let X; Y 2 I and jXj = jY j + 1. Then there exists x 2 X n Y such that Y [ fxg 2 I. X 2 I : an independent set of M. An independent set X of M is a base of M if it is a maximal independent set. Matroid E : set, I : a set of subsets of E. A pair M = (E; I) is a matroid on E if it satisfies the following three axioms. (T1) I= 6 ;. (T2) If X 2 I and Y ⊆ X, then Y 2 I. (T3) Let X; Y 2 I and jXj = jY j + 1. Then there exists x 2 X n Y such that Y [ fxg 2 I. X 2 I : an independent set of M. An independent set X of M is a base of M if it is a maximal independent set. Matroid E : set, I : a set of subsets of E. A pair M = (E; I) is a matroid on E if it satisfies the following three axioms. (T1) I= 6 ;. (T2) If X 2 I and Y ⊆ X, then Y 2 I. (T3) Let X; Y 2 I and jXj = jY j + 1. Then there exists x 2 X n Y such that Y [ fxg 2 I. X 2 I : an independent set of M. An independent set X of M is a base of M if it is a maximal independent set. Matroid E : set, I : a set of subsets of E. A pair M = (E; I) is a matroid on E if it satisfies the following three axioms. (T1) I= 6 ;. (T2) If X 2 I and Y ⊆ X, then Y 2 I. (T3) Let X; Y 2 I and jXj = jY j + 1. Then there exists x 2 X n Y such that Y [ fxg 2 I. X 2 I : an independent set of M. An independent set X of M is a base of M if it is a maximal independent set. Binary Matroid Let A be a V × W (0; 1)-matrix. The binary matroid M(A) of A is defined as (W; fU ⊆ W : U represent a linearly independent set of Ag). By the definition, a base of M(A) forms a maximally lin. independent set of column space of A. a b c d e 1 0 0 1 1! A = 0 1 0 1 1 0 0 1 0 1 fa; b; cg 2 I : a base. Graphic Matroid Let G be a graph. The graphic matroid M(G) of G is defined as M(G) = (E(G); fU ⊆ E(G): U forms a forest in Gg). By the definition, a base of M(G) forms a spanning tree of G. a e f G b d c fa; f; e; dg 2 I : a base Graphic Matroid 1 a e 2 f 5 G b d 3 c 4 a b c d e f 10 1 0 0 0 1 11 2B 1 1 0 0 0 0C I = B C G 3B 0 1 1 0 0 1C 4@ 0 0 1 1 0 0A 5 0 0 0 1 1 0 Graphic and Binary Matroid Theorem Let G be a graph and IG be the vertex-edge(V × E) incidence matrix of G. Then M(IG) = M(G). Therefore every graphic matroid is binary. 0. Problem. f f g G=e G=e n g Graph minor Let G = (V; E) be a graph. • G=e Contraction an edge e 2 E • G n e Deletion an edge e 2 E • G n v Deletion a vertex v 2 V f e g G f f g G=e G=e n g Graph minor Let G = (V; E) be a graph. • G=e Contraction an edge e 2 E • G n e Deletion an edge e 2 E • G n v Deletion a vertex v 2 V f e g G f f g G=e G=e n g Graph minor Let G = (V; E) be a graph. • G=e Contraction an edge e 2 E • G n e Deletion an edge e 2 E • G n v Deletion a vertex v 2 V f e g G f G=e n g Graph minor Let G = (V; E) be a graph. • G=e Contraction an edge e 2 E • G n e Deletion an edge e 2 E • G n v Deletion a vertex v 2 V f f e g g G G=e Graph minor Let G = (V; E) be a graph. • G=e Contraction an edge e 2 E • G n e Deletion an edge e 2 E • G n v Deletion a vertex v 2 V f f f e g g G G=e G=e n g No tree-width: 3 tree-width: 2 Graph minor Does the right graph has the left graph as a minor??? tree-width: 3 tree-width: 2 Graph minor Does the right graph has the left graph as a minor??? No Graph minor Does the right graph has the left graph as a minor??? No tree-width: 3 tree-width: 2 Pivot-minors of graphs We are interested in another operation. • G ^ uv Pivot-operation on an edge uv 2 E But in this talk, every graph has no triangle. C does not appear. Pivot-minors of graphs We are interested in another operation. • G ^ uv Pivot-operation on an edge uv 2 E But in this talk, every graph has no triangle. C does not appear. Pivot-minors of graphs We are interested in another operation. • G ^ uv Pivot-operation on an edge uv 2 E But in this talk, every graph has no triangle. C does not appear. Pivot-minors of graphs We are interested in another operation. • G ^ uv Pivot-operation on an edge uv 2 E But in this talk, every graph has no triangle. C does not appear. e e G G ^ e Pivot-minors of graphs For example, • H : a pivot-minor of a graph G if H is obtained from G by applying a sequence of pivoting edges and vertex deletions. e G ^ e Pivot-minors of graphs For example, e G • H : a pivot-minor of a graph G if H is obtained from G by applying a sequence of pivoting edges and vertex deletions. Pivot-minors of graphs For example, e e G G ^ e • H : a pivot-minor of a graph G if H is obtained from G by applying a sequence of pivoting edges and vertex deletions. Pivot-minors of graphs For example, e e G G ^ e • H : a pivot-minor of a graph G if H is obtained from G by applying a sequence of pivoting edges and vertex deletions. Pivot-minors of graphs For example, e e G G ^ e • H : a pivot-minor of a graph G if H is obtained from G by applying a sequence of pivoting edges and vertex deletions. v0 v2 v1 v3 v4 v5 v6 v7 v0 v2 v1 v4 v3 v5 v6 v7 v0 v2 v1 v4 v3 v6 v5 v7 v0 v2 v4 v6 v7 Example ^v1v2 ^ v3v4 ^ v5v6 n fv1; v3; v5g v0 v1 v2 v3 v4 v5 v6 v7 v0 v2 v1 v4 v3 v5 v6 v7 v0 v2 v1 v4 v3 v6 v5 v7 v0 v2 v4 v6 v7 Example ^v1v2 ^ v3v4 ^ v5v6 n fv1; v3; v5g v0 v1 v2 v3 v4 v5 v6 v7 v0 v2 v1 v3 v4 v5 v6 v7 v0 v2 v1 v4 v3 v6 v5 v7 v0 v2 v4 v6 v7 Example ^v1v2 ^ v3v4 ^ v5v6 n fv1; v3; v5g v0 v1 v2 v3 v4 v5 v6 v7 v0 v2 v1 v3 v4 v5 v6 v7 v0 v2 v1 v4 v3 v5 v6 v7 v0 v2 v4 v6 v7 Example ^v1v2 ^ v3v4 ^ v5v6 n fv1; v3; v5g v0 v1 v2 v3 v4 v5 v6 v7 v0 v2 v1 v3 v4 v5 v6 v7 v0 v2 v1 v4 v3 v5 v6 v7 v0 v2 v1 v4 v3 v6 v5 v7 Example ^v1v2 ^ v3v4 ^ v5v6 n fv1; v3; v5g v0 v1 v2 v3 v4 v5 v6 v7 v0 v2 v1 v3 v4 v5 v6 v7 v0 v2 v1 v4 v3 v5 v6 v7 v0 v2 v1 v4 v3 v6 v5 v7 v0 v2 v4 v6 v7 Complete binary tree The incidence graph of a complete binary tree rank-width: 1 rank-width: 1 k linear r-w: 2 linear rank-width: b 2 c + 1 Question Could the incidence graph of a complete binary tree have the B6 graph as a pivot-minor??? a b B6 graph k linear r-w: 2 linear rank-width: b 2 c + 1 Question Could the incidence graph of a complete binary tree have the B6 graph as a pivot-minor??? a b B6 graph rank-width: 1 rank-width: 1 Question Could the incidence graph of a complete binary tree have the B6 graph as a pivot-minor??? a b B6 graph rank-width: 1 rank-width: 1 k linear r-w: 2 linear rank-width: b 2 c + 1 We have two proofs of the theorem.
Load more

Recommended publications
  • Templates for Representable Matroids Kevin Manuel Grace Louisiana State University and Agricultural and Mechanical College, [email protected]
    Louisiana State University LSU Digital Commons LSU Doctoral Dissertations Graduate School 6-12-2018 Templates for Representable Matroids Kevin Manuel Grace Louisiana State University and Agricultural and Mechanical College, [email protected] Follow this and additional works at: https://digitalcommons.lsu.edu/gradschool_dissertations Part of the Discrete Mathematics and Combinatorics Commons Recommended Citation Grace, Kevin Manuel, "Templates for Representable Matroids" (2018). LSU Doctoral Dissertations. 4610. https://digitalcommons.lsu.edu/gradschool_dissertations/4610 This Dissertation is brought to you for free and open access by the Graduate School at LSU Digital Commons. It has been accepted for inclusion in LSU Doctoral Dissertations by an authorized graduate school editor of LSU Digital Commons. For more information, please [email protected]. TEMPLATES FOR REPRESENTABLE MATROIDS A Dissertation Submitted to the Graduate Faculty of the Louisiana State University and Agricultural and Mechanical College in partial fulfillment of the requirements for the degree of Doctor of Philosophy in The Department of Mathematics by Kevin Manuel Grace B.S., Pensacola Christian College, 2006 M.S., University of South Alabama, 2010 August 2018 \It is the glory of God to conceal a thing: but the honour of kings is to search out a matter." Proverbs 25:2 ii Acknowledgments It is a profound understatement to say that this dissertation would not have been possible without my advisor, Professor Stefan van Zwam. He has had a tremendous influence on every aspect of this dissertation. I would especially like to thank him for his guidance on presenting my research and for his help with the programming that was necessary. It was an honor to be his first graduate student.
    [Show full text]
  • On the Excluded Minors for Matroids of Branch-Width Three
    On the Excluded Minors for Matroids of Branch-Width Three Petr Hlinˇen´y∗ School of Mathematical and Computing Sciences, Victoria University, P.O. Box 600, Wellington, New Zealand; and Institute for Theoretical Computer Science†(ITI MFF), Charles University, Malostransk´en´am. 25, 118 00 Praha 1, Czech Republic. [email protected] Submitted: March 19, 2002; Accepted: July 23, 2002. Abstract Knowing the excluded minors for a minor-closed matroid property provides a useful alternative characterization of that property. It has been shown in [R. Hall, J. Oxley, C. Semple, G. Whittle, On Matroids of Branch-Width Three, submitted 2001] that if M is an excluded minor for matroids of branch-width 3, then M has at most 14 elements. We show that there are exactly 10 such binary matroids M (7 of which are regular), proving a conjecture formulated by Dharmatilake in 1994. We also construct numbers of such ternary and quaternary matroids M, and provide a simple practical algorithm for finding a width-3 branch-decomposition of a matroid. The arguments in our paper are computer-assisted | we use a program Macek [P. Hlinˇen´y, The Macek Program, http://www.mcs.vuw.ac.nz/research/macek, 2002] for structural computations with represented matroids. Unfortunately, it seems to be infeasible to search through all matroids on at most 14 elements. Keywords: representable matroid, minor, branch-width. MR Subject Classifications: 05B35, 05C83 ∗The research was supported by a New Zealand Marsden Fund research grant to Geoff Whittle. y(Supported by the Ministry of Education of Czech Republic as project LN00A056.) the electronic journal of combinatorics 9 (2002), #R32 1 1 Introduction We assume that the reader is familiar with basic terms of graph theory.
    [Show full text]
  • Well-Quasi-Ordering of Matrices Under Schur Complement And
    Well-Quasi-Ordering of Matrices under Schur Complement and Applications to Directed Graphs Mamadou Moustapha Kanté Clermont-Université, Université Blaise Pascal, LIMOS, CNRS Complexe Scientifique des Cézeaux 63173 Aubiére Cedex, France [email protected] Abstract In [Rank-Width and Well-Quasi-Ordering of Skew-Symmetric or Symmetric Matri- ces, arXiv:1007.3807v1] Oum proved that, for a fixed finite field F, any infinite se- quence M1, M2,... of (skew) symmetric matrices over F of bounded F-rank-width has a pair i < j, such that Mi is isomorphic to a principal submatrix of a principal pivot transform of Mj. We generalise this result to σ-symmetric matrices introduced by Rao and myself in [The Rank-Width of Edge-Coloured Graphs, arXiv:0709.1433v4]. (Skew) symmetric matrices are special cases of σ-symmetric matrices. As a by- product, we obtain that for every infinite sequence G1, G2,... of directed graphs of bounded rank-width there exist a pair i < j such that Gi is a pivot-minor of Gj. Another consequence is that non-singular principal submatrices of a σ-symmetric matrix form a delta-matroid. We extend in this way the notion of representability of delta-matroids by Bouchet. Key words: rank-width; sigma-symmetry; edge-coloured graph; well-quasi-ordering; principal pivot transform; pivot-minor. arXiv:1102.2134v3 [math.CO] 8 Jul 2014 1 Introduction Clique-width [6] is a graph complexity measure that emerges in the works by Courcelle et al. (see for instance the book [7]). It extends tree-width [21] in the sense that graph classes of bounded tree-width have bounded clique- width, but the converse is false (distance hereditary graphs have clique-width at most 3 and unbounded tree-width).
    [Show full text]
  • Graphs of Bounded Rank-Width
    Graphs of Bounded Rank-width Sang-il Oum A Dissertation Presented to the Faculty of Princeton University in Candidacy for the Degree of Doctor of Philosophy Recommended for Acceptance by the Program in Applied and Computational Mathematics May 2005 c Copyright by Sang-il Oum, 2005. All Rights Reserved iii Abstract We define rank-width of graphs to investigate clique-width. Rank-width is a com- plexity measure of decomposing a graph in a kind of tree-structure, called a rank- decomposition. We show that graphs have bounded rank-width if and only if they have bounded clique-width. It is unknown how to recognize graphs of clique-width at most k for fixed k > 3 in polynomial time. However, we find an algorithm recognizing graphs of rank-width at most k, by combining following three ingredients. First, we construct a polynomial-time algorithm, for fixed k, that confirms rank- width is larger than k or outputs a rank-decomposition of width at most f(k) for some function f. It was known that many hard graph problems have polynomial-time algo- rithms for graphs of bounded clique-width, however, requiring a given decomposition corresponding to clique-width (k-expression); we remove this requirement. Second, we define graph vertex-minors which generalizes matroid minors, and prove that if {G1,G2,...} is an infinite sequence of graphs of bounded rank-width, then there exist i < j such that Gi is isomorphic to a vertex-minor of Gj. Conse- quently there is a finite list Ck of graphs such that a graph has rank-width at most k if and only if none of its vertex-minors are isomorphic to a graph in Ck.
    [Show full text]
  • Matroids Graphs Give Isomorphic Matroid Structure?
    EECS 495: Combinatorial Optimization Lecture 7 Matroid Representation, Matroid Optimization Reading: Schrijver, Chapters 39 and 40 graph and S = E. A set F ⊆ E is indepen- dent if it is acyclic. Food for thought: can two non-isomorphic Matroids graphs give isomorphic matroid structure? Recap Representation Def: A matroid M = (S; I) is a finite ground Def: For a field F , a matroid M is repre- set S together with a collection of indepen- sentable over F if it is isomorphic to a linear dent sets I ⊆ 2S satisfying: matroid with matrix A and linear indepen- dence taken over F . • downward closed: if I 2 I and J ⊆ I, 2 then J 2 I, and Example: Is uniform matroid U4 binary? Need: matrix A with entries in f0; 1g s.t. no • exchange property: if I;J 2 I and jJj > column is the zero vector, no two rows sum jIj, then there exists an element z 2 J nI to zero over GF(2), any three rows sum to s.t. I [ fzg 2 I. GF(2). Def: A basis is a maximal independent set. • if so, can assume A is 2×4 with columns The cardinality of a basis is the rank of the 1/2 being (0; 1) and (1; 0) and remaining matroid. two vectors with entries in 0; 1 neither all k Def: Uniform matroids Un are given by jSj = zero. n, I = fI ⊆ S : jIj ≤ kg. • only three such non-zero vectors, so can't Def: Linear matroids: Let F be a field, A 2 have all pairs indep.
    [Show full text]
  • Arxiv:1910.05689V1 [Math.CO]
    ON GRAPHIC ELEMENTARY LIFTS OF GRAPHIC MATROIDS GANESH MUNDHE1, Y. M. BORSE2, AND K. V. DALVI3 Abstract. Zaslavsky introduced the concept of lifted-graphic matroid. For binary matroids, a binary elementary lift can be defined in terms of the splitting operation. In this paper, we give a method to get a forbidden-minor characterization for the class of graphic matroids whose all lifted-graphic matroids are also graphic using the splitting operation. Keywords: Elementary lifts; splitting; binary matroids; minors; graphic; lifted-graphic Mathematics Subject Classification: 05B35; 05C50; 05C83 1. Introduction For undefined notions and terminology, we refer to Oxley [13]. A matroid M is quotient of a matroid N if there is a matroid Q such that, for some X ⊂ E(Q), N = Q\X and M = Q/X. If |X| = 1, then M is an elementary quotient of N. A matroid N is a lift of M if M is a quotient of N. If M is an elementary quotient of N, then N is an elementary lift of M. A matroid N is a lifted-graphic matroid if there is a matroid Q with E(Q)= E(N) ∪ e such that Q\e = N and Q/e is graphic. The concept of lifted-graphic matroid was introduced by Zaslavsky [19]. Lifted-graphic matroids play an important role in the matroid minors project of Geelen, Gerards and Whittle [9, 10]. Lifted- graphic matroids are studied in [4, 5, 6, 7, 19]. This class is minor-closed. In [5], it is proved that there exist infinitely many pairwise non-isomorphic excluded minors for the class of lifted-graphic matroids.
    [Show full text]
  • Matroid Pathwidth and Code Trellis Complexity∗ 1
    MATROID PATHWIDTH AND CODE TRELLIS COMPLEXITY∗ NAVIN KASHYAP† Abstract. We relate the notion of matroid pathwidth to the minimum trellis state-complexity (which we term trellis-width) of a linear code, and to the pathwidth of a graph. By reducing from the problem of computing the pathwidth of a graph, we show that the problem of determining the pathwidth of a representable matroid is NP-hard. Consequently, the problem of computing the trellis-width of a linear code is also NP-hard. For a finite field F, we also consider the class of F-representable matroids of pathwidth at most w, and correspondingly, the family of linear codes over F with trellis-width at most w. These are easily seen to be minor-closed. Since these matroids (and codes) have branchwidth at most w, a result of Geelen and Whittle shows that such matroids (and the corresponding codes) are characterized by finitely many excluded minors. We provide the complete list of excluded minors for w = 1, and give a partial list for w = 2. Key words. Matroids, pathwidth, linear codes, trellis complexity, NP-hard. AMS subject classifications. 05B35, 94B05 1. Introduction. The notion of pathwidth of a matroid has received some recent attention in the matroid theory literature [7], [9]. This notion has long been studied in the coding theory literature, where it is used as a measure of trellis complexity of a linear code [14], [5], [17]. However, there appears to be no standard coding-theoretic nomenclature for this notion. It has been called the state complexity of a code in [10], but the use of this term there conflicts slightly with its use in [17].
    [Show full text]
  • Matroid Optimization and Algorithms Robert E. Bixby and William H
    Matroid Optimization and Algorithms Robert E. Bixby and William H. Cunningham June, 1990 TR90-15 MATROID OPTIMIZATION AND ALGORITHMS by Robert E. Bixby Rice University and William H. Cunningham Carleton University Contents 1. Introduction 2. Matroid Optimization 3. Applications of Matroid Intersection 4. Submodular Functions and Polymatroids 5. Submodular Flows and other General Models 6. Matroid Connectivity Algorithms 7. Recognition of Representability 8. Matroid Flows and Linear Programming 1 1. INTRODUCTION This chapter considers matroid theory from a constructive and algorithmic viewpoint. A substantial part of the developments in this direction have been motivated by opti­ mization. Matroid theory has led to a unification of fundamental ideas of combinatorial optimization as well as to the solution of significant open problems in the subject. In addition to its influence on this larger subject, matroid optimization is itself a beautiful part of matroid theory. The most basic optimizational property of matroids is that for any subset every max­ imal independent set contained in it is maximum. Alternatively, a trivial algorithm max­ imizes any {O, 1 }-valued weight function over the independent sets. Most of matroid op­ timization consists of attempts to solve successive generalizations of this problem. In one direction it is generalized to the problem of finding a largest common independent set of two matroids: the matroid intersection problem. This problem includes the matching problem for bipartite graphs, and several other combinatorial problems. In Edmonds' solu­ tion of it and the equivalent matroid partition problem, he introduced the notions of good characterization (intimately related to the NP class of problems) and matroid (oracle) algorithm.
    [Show full text]
  • The Circuit and Cocircuit Lattices of a Regular Matroid
    University of Vermont ScholarWorks @ UVM Graduate College Dissertations and Theses Dissertations and Theses 2020 The circuit and cocircuit lattices of a regular matroid Patrick Mullins University of Vermont Follow this and additional works at: https://scholarworks.uvm.edu/graddis Part of the Mathematics Commons Recommended Citation Mullins, Patrick, "The circuit and cocircuit lattices of a regular matroid" (2020). Graduate College Dissertations and Theses. 1234. https://scholarworks.uvm.edu/graddis/1234 This Thesis is brought to you for free and open access by the Dissertations and Theses at ScholarWorks @ UVM. It has been accepted for inclusion in Graduate College Dissertations and Theses by an authorized administrator of ScholarWorks @ UVM. For more information, please contact [email protected]. The Circuit and Cocircuit Lattices of a Regular Matroid A Thesis Presented by Patrick Mullins to The Faculty of the Graduate College of The University of Vermont In Partial Fulfillment of the Requirements for the Degree of Master of Science Specializing in Mathematical Sciences May, 2020 Defense Date: March 25th, 2020 Dissertation Examination Committee: Spencer Backman, Ph.D., Advisor Davis Darais, Ph.D., Chairperson Jonathan Sands, Ph.D. Cynthia J. Forehand, Ph.D., Dean of Graduate College Abstract A matroid abstracts the notions of dependence common to linear algebra, graph theory, and geometry. We show the equivalence of some of the various axiom systems which define a matroid and examine the concepts of matroid minors and duality before moving on to those matroids which can be represented by a matrix over any field, known as regular matroids. Placing an orientation on a regular matroid M allows us to define certain lattices (discrete groups) associated to M.
    [Show full text]
  • Binary Matroid Whose Bases Form a Graphic Delta-Matroid
    Characterizing matroids whose bases form graphic delta-matroids Duksang Lee∗2,1 and Sang-il Oum∗1,2 1Discrete Mathematics Group, Institute for Basic Science (IBS), Daejeon, South Korea 2Department of Mathematical Sciences, KAIST, Daejeon, South Korea Email: [email protected], [email protected] September 1, 2021 Abstract We introduce delta-graphic matroids, which are matroids whose bases form graphic delta- matroids. The class of delta-graphic matroids contains graphic matroids as well as cographic matroids and is a proper subclass of the class of regular matroids. We give a structural charac- terization of the class of delta-graphic matroids. We also show that every forbidden minor for the class of delta-graphic matroids has at most 48 elements. 1 Introduction Bouchet [2] introduced delta-matroids which are set systems admitting a certain exchange axiom, generalizing matroids. Oum [18] introduced graphic delta-matroids as minors of binary delta-matroids having line graphs as their fundamental graphs and proved that bases of graphic matroids form graphic delta-matroids. We introduce delta-graphic matroids as matroids whose bases form a graphic delta- matroid. Since the class of delta-graphic matroids is closed under taking dual and minors, it contains both graphic matroids and cographic matroids. Alternatively one can define delta-graphic matroids as binary matroids whose fundamental graphs are pivot-minors of line graphs. See Section 2 for the definition of pivot-minors. Our first theorem provides a structural characterization of delta-graphic matroids. A wheel graph W is a graph having a vertex s adjacent to all other vertices such that W \s is a cycle.
    [Show full text]
  • Ideal Binary Clutters, Connectivity and a Conjecture of Seymour
    IDEAL BINARY CLUTTERS, CONNECTIVITY, AND A CONJECTURE OF SEYMOUR GERARD´ CORNUEJOLS´ AND BERTRAND GUENIN ABSTRACT. A binary clutter is the family of odd circuits of a binary matroid, that is, the family of circuits that ¼; ½ intersect with odd cardinality a fixed given subset of elements. Let A denote the matrix whose rows are the Ü ¼ : AÜ ½g characteristic vectors of the odd circuits. A binary clutter is ideal if the polyhedron f is ×Ø Ì Ì integral. Examples of ideal binary clutters are ×Ø-paths, -cuts, -joins or -cuts in graphs, and odd circuits in weakly bipartite graphs. In 1977, Seymour conjectured that a binary clutter is ideal if and only if it does not µ Ä Ç b´Ç à à contain F , ,or as a minor. In this paper, we show that a binary clutter is ideal if it does not contain 7 5 5 five specified minors, namely the three above minors plus two others. This generalizes Guenin’s characterization Ì of weakly bipartite graphs, as well as the theorem of Edmonds and Johnson on Ì -joins and -cuts. 1. INTRODUCTION E ´Àµ A clutter À is a finite family of sets, over some finite ground set , with the property that no set of jE ´Àµj À fÜ ¾ Ê : À contains, or is equal to, another set of . A clutter is said to be ideal if the polyhedron · È Ü ½; 8Ë ¾Àg ¼; ½ i is an integral polyhedron, that is, all its extreme points have coordinates. A i¾Ë À = ; À = f;g À A´Àµ clutter À is trivial if or .
    [Show full text]
  • EULERIAN and BIPARTITE ORIENTABLE MATROIDS Laura E
    chapter02 OUP012/McDiarmid (Typeset by SPi, Delhi) 11 of 27 July 4, 2006 13:12 2 EULERIAN AND BIPARTITE ORIENTABLE MATROIDS Laura E. Ch´avez Lomel´ı and Luis A. Goddyn Welsh [6] extended to the class of binary matroids a well-known theorem regarding Eulerian graphs. Theorem 2.1 Let M be a binary matroid. The ground set E(M) can be partitioned into circuits if and only if every cocircuit of M has even cardinality. Further work of Brylawski and Heron (see [4, p. 315]) explores other char- acterizations of Eulerian binary matroids. They showed, independently, that a binary matroid M is Eulerian if and only if its dual, M ∗, is a binary affine matroid. More recently, Shikare and Raghunathan [5] have shown that a binary matroid M is Eulerian if and only if the number of independent sets of M is odd. This chapter is concerned with extending characterizations of Eulerian graphs via orientations. An Eulerian tour of a graph G induces an orientation with the property that every cocircuit (minimal edge cut) in G is traversed an equal number of times in each direction. In this sense, we can say that the orientation is balanced. Applying duality to planar graphs, these notions produce characterizations of bipartite graphs. Indeed the notions of flows and colourings of regular matroids can be formulated in terms of orientations, as was observed by Goddyn et al. [2]. The equivalent connection for graphs had been made by Minty [3]. In this chapter, we further extend these notions to oriented matroids. Infor- mally, an oriented matroid is a matroid together with additional sign informa- tion.
    [Show full text]