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List of Symbols List of Symbols A(D) the set of arcs of D ; 49 B(G) the bipartite graph of graph G; 299 c the capacity function of a network; 87 c(a) the capacity of arc a; 87 en the cycle of length n ; 19 capK the sum of the capacities of the arcs in K; 89 cl(G) the closure of G ; 172 dG(v) the degree of the vertex v in G; 14 d(v) the degree of the vertex v in a graph ; 14 (dl , d- , . .. , dn ) the degree sequence of a graph ; 15 d(u, v) the length of a shortest u- v path (respectively directed path) in a graph; 20 (respectively digraph ; 64) d"i>(v) the outdegree of v in D; 50 d+(v) the outdegree of v in a digraph; 50 di)(v) the indegree of v in D; 51 d-(v) the indegree of v in a digraph; 51 dD(v) the degree of v in D; 51 D a directed graph or digraph; 49 Dn the dihedral group of order 2n; 46 diam(G ) the diameter of G; 47 E(G) the edge set of G; 3 E the edge set of a graph; 5 e(v) the eccentricity of vertex v; 110 f +(5) f([5 , 5]), where 55; V(D); 88 214 List of Symbols f-(S) f([5, S]), where S ~ V(D); 88 f(G) the number of faces of a planar graph G ; 244 f(G ;A) the chromatic polynomial of G; 229 t; f«u, v)); 88 G a graph ; 3 G C the complement of a simple graph G; 9 G(D) the underlying graph of D; 49 G(X, Y) a bipartite graph G with bipartition (X, Y); 9 G[S] the subgraph of G induced by the subset S of V(G); 11 G[E'] the subgraph of G induced by the subset E' of E(G); 11 G +uv the supergraph of G obtained by adding the new edge uv; 11 G-v the subgraph of G obtained by deleting the vertex v; 13 G-S the subgraph of G obtained by the deletion of the vertices in S; 13 G -e the subgraph of G obtained by deleting the edge e; 13 G-E' the subgraph of G obtained by the deletion of the edges in E'; 13 G) UGz the union of the two graphs G) and Gz ; 37 G( +Gz the sum of the two graphs Gland G2; 37 G) nGz the intersection ofthe two graphs G) and G2; 37 G( vGz the join of the two graphs G( and G2; 37 G] x Gz the Cartesian product of the graph G) with the graph G2; 38 the composition or lexicographic product of the graph G) with the graph G z; 39 the normal product or the strong product of the graph G) with the graph Gz; 40 the tensor product or the Kronecker product of the graph G 1 with the graph G 2 ; 41 G oe the graph obtained from G by contracting the edgee; 116 G* the canonical dual of the plane graph G; 256 G4 Grotszch graph; 213 Gk the k-th power of G; 42 ID the incidence map of D ; 49 IG the incidence map of G ; 3 Kn the complete graph on n vertices; 7 Kp,q the complete bipartite graph with part sizes p and q; 9 List of Symbols 215 K1,q the star of size q ; 9 K(G) the clique graph of G ; 299 L(G) the line graph of the graph G; 29 m(G) the size of G = the number of edges in G; 5 m the size (= the number ofedges) of a graph; 5 NG(v) the open neighborhood of the vertex V in G; 4 N(v) the open neighborhood of the vertex v in a graph; 4 Na[v] the closed neighborhood of the vertex v in G ; 4 N[v] the closed neighborhood of the vertex v in a graph; 4 n(G) the order of G = the number of vertices of G ;5 n the order of a graph; 5 NZ(v) the set of outneighbors of v in D; 50 N+(v) the set ofoutneighbors of v in a digraph; 50 N;(v) the set of inneighbors of v in D ; 50 N-(v) the set of inneighbors of v in a digraph; 50 N a network; 87 N(S) the neighbor set of S in a graph; 139 o(G) the number of odd components of G; 144 p the Petersen graph; 8 Pn the path on n vertices; 19 p-l the inverse of the path P; 19 Qn the n-cube; 136 rG the sum of r copies of the graph G ; 37 r(G) the radius of graph G; III S the symmetric group of degree n; 27 Sn the source of a network; 87 [S, S'] the set of all arcs having their tails in Sand heads in S' in the case of directed graphs; 55 (the set of all edges having one end in Sand the other end in S' in the case of undirected graphs; 67) S(v) the score of the vertex v in a tournament; 61 (s(vd, S(V2), •••, S(Vn » the score vector of a tournament with vertex set {VI, V2, ..•, vn}; 61 the sink of a network; 87 vOel VIe2v2 '.. e; u, a (vo, vr ) walks in a graph; 18 V the vertex set of a graph; 5 V(D) the set of vertices of D;49 V(G) the vertex set of G ; 3 val f the value of the flow f = f+(s) - f-(s) = f-(t) - f+(/) ; 89 en v KI, the wheel with n spokes; 37 216 List of Symbols = is isomorphic to; 7 a (G ) the independence number of G; 129 a'(G) the cardinality of a maximum matching of G; 130 fJ(G) the covering number of G; 129 fJ'(G) the cardinality of a minimum edge covering of G; 130 r eG) the group of automorphisms of the graph G; 25 8(G) the minimum degree of G ; 14 8 the minimum degree of a graph; 14 ~ (G) the maximum degree of G; 14 ~ the maximum degree of a graph; 14 cPI °cP2 the composition of the mappings cPt and cP2 (cP2 followed by cPl); 26 A(G) the edge connectivity of G; 73 A the edge connectivity of a graph; 73 AcG the cyclical edge connectivity of G; 85 K(G) the vertex connectivity of G; 73 K the vertex connectivity of a graph; 73 B(G) the clique covering number of G = the min imum number of cliq ues of G that co ver the vertex set of G; 285 r eG) the number of spanning trees of G ; 116 w(G) the clique number of G = the order of a maximum clique of G; 285 w(G) the number of components of G; 20 x(G) the chromatic number of G; 199 x'(G) the edge chromatic number or chromatic index of G ; 2 15 References [1] A. V. Aho, J. E. Hopcroft, and J. D. Ullman , The Design and Analysis ofComputer Algorithms, Addi son-Wesley, Reading, Mass., 174. [2] K. Appel and W. Haken , Every planar map is four colorable: Part I-Discharging, Illinois J. Math . 21 (1977),429-490. [3] K. Appel and W. Haken, The solution of the four-color-map problem, Scientific American 237(4) (1977), 108-121. [4] K. Appel , W. Haken, and J. Koch, Every planar map is four colorable: Part 11­ Reducibility, Illinois J. Math. 21 (1977),491-567. [5] R. Aravamudhan and B. Rajendran (personal communication). [6] R. Balakrishnan and P. Paulraja, Powers of chordal graphs, J. Austral. Math. Soc. Ser. A 35 (1983), 211-217 . [7] R. Balakrishnan and P. Paulraja, Line graphs ofsubdivision graphs, J. Combin. Info. and Sys. Sci. 10 (1985), 33-35 . [8] R. Balakrishnan and P. Paulraja, Chordal graphs and some of their derived graphs, Congressus Numerantium, 53 (1986),33-35. [9] M. Behzad, G. Chartrand, and L. Lesniak-Foster, Graphs and Digraphs, Prindle, Weber & Schmidt Int. Series, Boston, Mass., 1979. [10] L. W. Beineke, On derived graphs and digraphs, in Beitrdge zur Graphentheorie, eds., H. Sachs, H. J. Voss, and H. Walther, Teubner, Leipzig, 1968, 17-23. [11] S. Benzer, On the topology of the genetic fine structure, Proc. Nat. Acad. Sci., USA, 45 (1959), 1607-1620. [12] C. Berge, Farbung von Graphen, deren sarntliche bzw. deren ungerade Kreise starr sind, Wiss Z Martin-Luther-Univ. Halle-Wittenberg Math. Natur. Reihe (1961), 114-115. 218 References [13J C. Berge, Graphs and Hypergraphs, North-Holland, London , 1973. [14] C. Berge and V. Chvatal , Topics on perfect graphs, Ann. Discrete Math . 21 (1984) . [15] C. Berge, Graphs, Elsevier Science Publishers B.Y., North-Holland, London, 1991. [16] G. Birkhoff and D. Lewis, Chromatic polynomials, Trans. Amer. Math. Soc. 60 (1946), 355-451. [17J J. A. Bondy, Pancyclic graphs , J. Combin. Theory Ser. B 11 (1971), 80-84. [18J J. A. 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