On Switching to H‐Free Graphs

In this article, we study the problem of deciding if, for a fixed graph H, a given graph is switching equivalent to an H‐free graph. Polynomial‐time algorithms are known for H having at most three vertices or isomorphic to P₄. We show that for H isomorphic to a claw, the problem is polynomial, too. On the other hand, we give infinitely many graphs H such that the problem is NP‐complete, thus solving an open problem [Kratochvíl, Ne?et?il and Zýka, Ann Discrete Math 51 (1992)]. Further, we give a characterization of graphs switching equivalent to a K_{1, 2}‐free graph by ten forbidden‐induced subgraphs, each having five vertices. We also give the forbidden‐induced subgraphs for graphs switching equivalent to a forest of bounded vertex degrees.     

Partial characterizations of circular‐arc graphs

A circular‐arc graph is the intersection graph of a family of arcs on a circle. A characterization by forbidden induced subgraphs for this class of graphs is not known, and in this work we present a partial result in this direction. We characterize circular‐arc graphs by a list of minimal forbidden induced subgraphs when the graph belongs to any of the following classes: P₄ ‐free graphs, paw‐free graphs, claw‐free chordal graphs and diamond‐free graphs. © 2009 Wiley Periodicals, Inc. J Graph Theory 61: 289–306, 2009     

Restricted unimodular chordal graphs

A chordal graph is called restricted unimodular if each cycle of its vertex‐clique incidence bipartite graph has length divisible by 4. We characterize these graphs within all chordal graphs by forbidden induced subgraphs, by minimal relative separators, and in other ways. We show how to construct them by starting from block graphs and multiplying vertices subject to a certain restriction, which leads to a linear‐time recognition algorithm. We show how they are related to other classes such as distance‐hereditary chordal graphs and strongly chordal graphs. © 1999 John Wiley & Sons, Inc. J Graph Theory 30: 121–136, 1999     

Characterizing Flag Graphs and Induced Subgraphs of Cartesian Product Graphs

The vertices of the flag graph Φ(P) of a graded poset P are its maximal chains. Two vertices are adjacent whenever two maximal chains differ in exactly one element. In this paper we characterize induced subgraphs of Cartesian product graphs and flag graphs of graded posets. The latter class of graphs lies between isometric and induced subgraphs of Cartesian products in the embedding structure theory. Both characterization use certain edge-labelings of graphs.     

Characterizing subgraphs of Hamming graphs

Cartesian products of complete graphs are known as Hamming graphs. Using embeddings into Cartesian products of quotient graphs we characterize subgraphs, induced subgraphs, and isometric subgraphs of Hamming graphs. For instance, a graph G is an induced subgraph of a Hamming graph if and only if there exists a labeling of E(G) fulfilling the following two conditions: (i) edges of a triangle receive the same label; (ii) for any vertices u and v at distance at least two, there exist two labels which both appear on any induced u, υ‐path. © 2005 Wiley Periodicals, Inc. J Graph Theory 49: 302–312, 2005     

The superregular graphs

A regular graph is superregular if it has no vertices or if the subgraphs induced by the neighbors and by the nonneighbors of each vertex are superregular. The superregular graphs are precisely the disjoint union of m isomorphic cliques, the Cartesian product of two isomorphic cliques, the five-cycle, and the complements of these graphs. © 1996 John Wiley & Sons, Inc.     

Partial characterizations of clique-perfect graphs I: Subclasses of claw-free graphs

A clique-transversal of a graph G is a subset of vertices that meets all the cliques of G. A clique-independent set is a collection of pairwise vertex-disjoint cliques. The clique-transversal number and clique-independence number of G are the sizes of a minimum clique-transversal and a maximum clique-independent set of G, respectively. A graph G is clique-perfect if these two numbers are equal for every induced subgraph of G. The list of minimal forbidden induced subgraphs for the class of clique-perfect graphs is not known. In this paper, we present a partial result in this direction; that is, we characterize clique-perfect graphs by a restricted list of forbidden induced subgraphs when the graph belongs to two different subclasses of claw-free graphs.     

Spanning subgraphs of graphs partitioned into two isomorphic pieces

A graph has the neighbor‐closed‐co‐neighbor, or ncc property, if for each of its vertices x, the subgraph induced by the neighbor set of x is isomorphic to the subgraph induced by the closed non‐neighbor set of x. As proved by Bonato and Nowakowski [ 5 ], graphs with the ncc property are characterized by the existence of perfect matchings satisfying certain local conditions. In the present article, we investigate the spanning subgraphs of ncc graphs, which we name sub‐ncc. Several equivalent characterizations of finite sub‐ncc graphs are given, along with a polynomial time algorithm for their recognition. The infinite sub‐ncc graphs are characterized, and we demonstrate the existence of a countable universal sub‐ncc graph satisfying a strong symmetry condition called pseudo‐homogeneity. © 2005 Wiley Periodicals, Inc. J Graph Theory     

Large classes of infinite k‐cop‐win graphs

While finite cop‐win finite graphs possess a good structural characterization, none is known for infinite cop‐win graphs. As evidence that such a characterization might not exist, we provide as large as possible classes of infinite graphs with finite cop number. More precisely, for each infinite cardinal κ and each positive integer k, we construct 2^κ non‐isomorphic k‐cop‐win graphs satisfying additional properties such as vertex‐transitivity, or having universal endomorphism monoid and automorphism group. © 2010 Wiley Periodicals, Inc. J Graph Theory 65: 334–342, 2010     

Claw-free graphs are edge reconstructible

The Edge Reconstruction Conjecture states that all graphs with at least four edges are determined by their edge-deleted subgraphs. We prove that this is true for claw-free graphs, those graphs with no induced subgraph isomorphic to K_1,3. This includes line graphs as a special case.     

On minimal forbidden subgraph characterizations of balanced graphs

A graph is balanced if its clique-matrix contains no edge–vertex incidence matrix of an odd chordless cycle as a submatrix. While a forbidden induced subgraph characterization of balanced graphs is known, there is no such characterization by minimal forbidden induced subgraphs. In this work, we provide minimal forbidden induced subgraph characterizations of balanced graphs restricted to graphs that belong to one of the following graph classes: complements of bipartite graphs, line graphs of multigraphs, and complements of line graphs of multigraphs. These characterizations lead to linear-time recognition algorithms for balanced graphs within the same three graph classes.     

Polar cographs

Polar graphs are a natural extension of some classes of graphs like bipartite graphs, split graphs and complements of bipartite graphs. A graph is (s,k)-polar if there exists a partition A,B of its vertex set such that A induces a complete s-partite graph (i.e., a collection of at most s disjoint stable sets with complete links between all sets) and B a disjoint union of at most k cliques (i.e., the complement of a complete k-partite graph).Recognizing a polar graph is known to be NP-complete. These graphs have not been extensively studied and no good characterization is known. Here we consider the class of polar graphs which are also cographs (graphs without induced path on four vertices). We provide a characterization in terms of forbidden subgraphs. Besides, we give an algorithm in time O(n) for finding a largest induced polar subgraph in cographs; this also serves as a polar cograph recognition algorithm. We examine also the monopolar cographs which are the (s,k)-polar cographs where min(s,k)?1. A characterization of these graphs by forbidden subgraphs is given. Some open questions related to polarity are discussed.     

Partial characterizations of coordinated graphs: line graphs and complements of forests

A graph G is coordinated if the minimum number of colors that can be assigned to the cliques of H in such a way that no two cliques with non-empty intersection receive the same color is equal to the maximum number of cliques of H with a common vertex, for every induced subgraph H of G. Coordinated graphs are a subclass of perfect graphs. The list of minimal forbidden induced subgraphs for the class of coordinated graphs is not known. In this paper, we present a partial result in this direction, that is, we characterize coordinated graphs by minimal forbidden induced subgraphs when the graph is either a line graph, or the complement of a forest. F. Bonomo, F. Soulignac, and G. Sueiro’s research partially supported by UBACyT Grant X184 (Argentina), and CNPq under PROSUL project Proc. 490333/2004-4 (Brazil). The research of G. Durán is partially supported by FONDECyT Grant 1080286 and Millennium Science Institute “Complex Engineering Systems” (Chile), and CNPq under PROSUL project Proc. 490333/2004-4 (Brazil).     

New ‐Edge‐Balanced Graphs from Bipartite Graphs

Let G be a graph of order n satisfying that there exists for which every graph of order n and size t is contained in exactly λ distinct subgraphs of the complete graph isomorphic to G. Then G is called t‐edge‐balanced and λ the index of G. In this article, new examples of 2‐edge‐balanced graphs are constructed from bipartite graphs and some further methods are introduced to obtain more from old.     

Locally homogeneous graphs from groups

A graph is called locally homogeneous if the subgraphs induced at any two points are isomorphic. in this Note we give a method for constructing locally homogeneous graphs from groups. the graphs constructable in this way are exactly the locally homogeneous graphs with a point symmetric universal cover. As an example we characterize the graphs that are locally n-cycles.     

Partial characterizations of clique-perfect graphs II: Diamond-free and Helly circular-arc graphs

A clique-transversal of a graph G is a subset of vertices that meets all the cliques of G. A clique-independent set is a collection of pairwise vertex-disjoint cliques. A graph G is clique-perfect if the sizes of a minimum clique-transversal and a maximum clique-independent set are equal for every induced subgraph of G. The list of minimal forbidden induced subgraphs for the class of clique-perfect graphs is not known. Another open question concerning clique-perfect graphs is the complexity of the recognition problem. Recently we were able to characterize clique-perfect graphs by a restricted list of forbidden induced subgraphs when the graph belongs to two different subclasses of claw-free graphs. These characterizations lead to polynomial time recognition of clique-perfect graphs in these classes of graphs. In this paper we solve the characterization problem in two new classes of graphs: diamond-free and Helly circular-arc () graphs. This last characterization leads to a polynomial time recognition algorithm for clique-perfect graphs.     

Planar embeddings with infinite faces

We investigate vertex‐transitive graphs that admit planar embeddings having infinite faces, i.e., faces whose boundary is a double ray. In the case of graphs with connectivity exactly 2, we present examples wherein no face is finite. In particular, the planar embeddings of the Cartesian product of the r‐valent tree with K₂ are comprehensively studied and enumerated, as are the automorphisms of the resulting maps, and it is shown for r = 3 that no vertex‐transitive group of graph automorphisms is extendable to a group of homeomorphisms of the plane. We present all known families of infinite, locally finite, vertex‐transitive graphs of connectivity 3 and an infinite family of 4‐connected graphs that admit planar embeddings wherein each vertex is incident with an infinite face. © 2003 Wiley Periodicals, Inc. J Graph Theory 42: 257–275, 2003     

On a paper of lake about infinite graphs

Lake has constructed graphs G and H such that H contains n disjoint subgraphs isomorphic to G for each positive integer n but H does not contain infinitely many disjoint subgraphs isomorphic to G. Here a concrete example of such graphs with H the lattice points of the plane is given.     

A generalization of line graphs: (X,Y)-intersection graphs

Given a pair (X, Y) of fixed graphs X and Y, the (X, Y)-intersection graph of a graph G is a graph whose vertices correspond to distinct induced subgraphs of G that are isomorphic to Y, and where two vertices are adjacent iff the intersection of their corresponding subgraphs contains an induced subgraph isomorphic to X. This generalizes the notion of line graphs, since the line graph of G is precisely the (K₁, K₂)-intersection graph of G. In this paper, we consider the forbidden induced subgraph characterization of (X, Y)-intersection graphs for various (X, Y) pairs; such consideration is motivated by the characterization of line graphs through forbidden induced subgraphs. For this purpose, we restrict our attention to hereditary pairs (a pair (X, Y) is hereditary if every induced subgraph of any (X, Y)-intersection graph is also an (X, Y)-intersection graph), since only for such pairs do (X, Y)-intersection graphs have forbidden induced subgraph characterizations. We show that for hereditary 2-pairs (a pair (X, Y) is a 2-pair if Y contains exactly two induced subgraphs isomorphic to X), the family of line graphs of multigraphs and the family of line graphs of bipartite graphs are the maximum and minimum elements, respectively, of the poset on all families of (X, Y)-intersection graphs ordered by set inclusion. We characterize 2-pairs for which the family of (X, Y)-intersection graphs are exactly the family of line graphs or the family of line graphs of multigraphs. © 1996 John Wiley & Sons, Inc.     

A Characterization of Domination Reducible Graphs

We introduce a hereditary class of domination reducible graphs where the minimum dominating set problem is polynomially solvable, and characterize this class in terms of forbidden induced subgraphs.Acknowledgments The research was supported by DIMACS 2002 and 2003 Awards. The author would like to thank the both referees for their valuable suggestions.Final version received: October 3, 2003