On classes of minimal circular-imperfect graphs

Circular-perfect graphs form a natural superclass of perfect graphs: on the one hand due to their definition by means of a more general coloring concept, on the other hand as an important class of χ-bound graphs with the smallest non-trivial χ-binding function χ(G)?ω(G)+1.The Strong Perfect Graph Conjecture, recently settled by Chudnovsky et al. [The strong perfect graph theorem, Ann. of Math. 164 (2006) 51-229], provides a characterization of perfect graphs by means of forbidden subgraphs. It is, therefore, natural to ask for an analogous conjecture for circular-perfect graphs, that is for a characterization of all minimal circular-imperfect graphs.At present, not many minimal circular-imperfect graphs are known. This paper studies the circular-(im)perfection of some families of graphs: normalized circular cliques, partitionable graphs, planar graphs, and complete joins. We thereby exhibit classes of minimal circular-imperfect graphs, namely, certain partitionable webs, a subclass of planar graphs, and odd wheels and odd antiwheels. As those classes appear to be very different from a structural point of view, we infer that formulating an appropriate conjecture for circular-perfect graphs, as analogue to the Strong Perfect Graph Theorem, seems to be difficult.     

Circular-imperfection of triangle-free graphs

Circular-perfect graphs form a natural superclass of the well-known perfect graphs by means of a more general coloring concept.For perfect graphs, a characterization by means of forbidden subgraphs was recently settled by Chudnovsky et al. [Chudnovsky, M., N. Robertson, P. Seymour, and R. Thomas, The Strong Perfect Graph Theorem, Annals of Mathematics 164 (2006) 51–229]. It is, therefore, natural to ask for an analogous characterization for circular-perfect graphs or, equivalently, for a characterization of all minimally circular-imperfect graphs.Our focus is the circular-(im)perfection of triangle-free graphs. We exhibit several different new infinite families of minimally circular-imperfect triangle-free graphs. This shows that a characterization of circular-perfect graphs by means of forbidden subgraphs is a difficult task, even if restricted to the class of triangle-free graphs. This is in contrary to the perfect case where it is long-time known that the only minimally imperfect triangle-free graphs are the odd holes [Tucker, A., Critical Perfect Graphs and Perfect 3-chromatic Graphs, J. Combin. Theory (B) 23 (1977) 143–149].     

Partial characterizations of clique-perfect and coordinated graphs: superclasses of triangle-free graphs

A graph is clique-perfect if the cardinality of a maximum clique-independent set equals the cardinality of a minimum clique-transversal, for all its induced subgraphs. A graph G is coordinated if the chromatic number of the clique graph of H equals 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 complete lists of minimal forbidden induced subgraphs for the classes of cliqueperfect and coordinated graphs are not known, but some partial characterizations have been obtained. In this paper, we characterize clique-perfect and coordinated graphs by minimal forbidden induced subgraphs when the graph is either paw-free or {gem,W₄,bull}-free, two superclasses of triangle-free graphs.     

Clique-perfectness of complements of line graphs

The clique-transversal number τ_c(G) of a graph G is the minimum size of a set of vertices meeting all the cliques. The clique-independence number α_c(G) of G is the maximum size of a collection of vertex-disjoint cliques. A graph is clique-perfect if these two numbers are equal for every induced subgraph of G. Unlike perfect graphs, the class of clique-perfect graphs is not closed under graph complementation nor is a characterization by forbidden induced subgraphs known. Nevertheless, partial results in this direction have been obtained. For instance, in [Bonomo, F., M. Chudnovsky and G. Durán, Partial characterizations of clique-perfect graphs I: Subclasses of claw-free graphs, Discrete Appl. Math. 156 (2008), pp. 1058–1082], a characterization of those line graphs that are clique-perfect is given in terms of minimal forbidden induced subgraphs. Our main result is a characterization of those complements of line graphs that are clique-perfect, also by means of minimal forbidden induced subgraphs. This implies an O(n²) time algorithm for deciding the clique-perfectness of complements of line graphs and, for those that are clique-perfect, finding α_c and τ_c.     

Constructions for normal graphs and some consequences

Normal graphs can be considered as weaker perfect graphs in several ways. However, only few graphs are known yet to be normal, apart from perfect graphs, odd holes, and odd antiholes of length ≥ 9. Körner and de Simone [J. Körner, C. de Simone, On the odd cycles of normal graphs, Discrete Appl. Math. 94 (1999) 161-169] conjectured that every ()-free graph is normal. As there exist normal graphs containing C₅, C₇, or , it is worth looking for other ways to construct or detect normal graphs. For that, we treat the behavior of normal graphs under certain construction techniques (substitution, composition, and clique identification), providing several ways to construct new normal graphs from normal and even not normal ones, and consider the corresponding structural decompositions (homogeneous sets, skew partitions, and clique cutsets). Our results imply that normal graphs cannot be characterized by means of decomposition techniques as well as by forbidden subgraphs. We address negative consequences for the algorithmic behavior of normal graphs, reflected by the fact that neither the imperfection ratio can be bounded for normal graphs nor a χ-binding function exists. The latter is even true for the class of ()-free graphs and related classes. We conclude that normal graphs are indeed only “normal”.     

Meyniel graphs are strongly perfect

Meyniel (Discrete Math.16 (1976), 339–342) proved that a graph is perfect whenever each of its odd cycles of length at least five has at least two chords. This result is strengthened by proving that every graph satisfying Meyniel's condition is strongly perfect (i.e., each of its induced subgraphs H contains a stable set which meets all the maximal cliques in H).     

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.     

Computing clique and chromatic number of circular-perfect graphs in polynomial time

A main result of combinatorial optimization is that clique and chromatic number of a perfect graph are computable in polynomial time (Grötschel et al. in Combinatorica 1(2):169–197, 1981). Perfect graphs have the key property that clique and chromatic number coincide for all induced subgraphs; we address the question whether the algorithmic results for perfect graphs can be extended to graph classes where the chromatic number of all members is bounded by the clique number plus one. We consider a well-studied superclass of perfect graphs satisfying this property, the circular-perfect graphs, and show that for such graphs both clique and chromatic number are computable in polynomial time as well. In addition, we discuss the polynomial time computability of further graph parameters for certain subclasses of circular-perfect graphs. All the results strongly rely upon Lovász’s Theta function.     

Partial characterizations of clique-perfect and coordinated graphs: Superclasses of triangle-free graphs

A graph G is clique-perfect if the cardinality of a maximum clique-independent set of H equals the cardinality of a minimum clique-transversal of H, for every induced subgraph H of G. 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 equals 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 complete lists of minimal forbidden induced subgraphs for the classes of clique-perfect and coordinated graphs are not known, but some partial characterizations have been obtained. In this paper, we characterize clique-perfect and coordinated graphs by minimal forbidden induced subgraphs when the graph is either paw-free or {gem, W₄, bull}-free, both superclasses of triangle-free graphs.     

Partial characterizations of circle graphs

A circle graph is the intersection graph of a family of chords on a circle. There is no known characterization of circle graphs by forbidden induced subgraphs that do not involve the notions of local equivalence or pivoting operations. We characterize circle graphs by a list of minimal forbidden induced subgraphs when the graph belongs to one of the following classes: linear domino graphs, P₄-tidy graphs, and tree-cographs. We also completely characterize by minimal forbidden induced subgraphs the class of unit Helly circle graphs, which are those circle graphs having a model whose chords have all the same length, are pairwise different, and satisfy the Helly property.     

Forbidden subgraphs that imply 2-factors

The connected forbidden subgraphs and pairs of connected forbidden subgraphs that imply a 2-connected graph is hamiltonian have been characterized by Bedrossian [Forbidden subgraph and minimum degree conditions for hamiltonicity, Ph.D. Thesis, Memphis State University, 1991], and extensions of these excluding graphs for general graphs of order at least 10 were proved by Faudree and Gould [Characterizing forbidden pairs for Hamiltonian properties, Discrete Math. 173 (1997) 45-60]. In this paper a complete characterization of connected forbidden subgraphs and pairs of connected forbidden subgraphs that imply a 2-connected graph of order at least 10 has a 2-factor will be proved. In particular it will be shown that the characterization for 2-factors is very similar to that for hamiltonian cycles, except there are seven additional pairs. In the case of graphs of all possible orders, there are four additional forbidden pairs not in the hamiltonian characterization, but a claw is part of each pair.     

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 the following classes: diamond-free graphs, P₄-free graphs, paw-free graphs, and claw-free chordal graphs.     

On minimal forbidden subgraph characterizations of balanced graphs

A {0, 1}-matrix is balanced if it contains no square submatrix of odd order with exactly two 1's per row and per column. Balanced matrices lead to ideal formulations for both set packing and set covering problems. Balanced graphs are those graphs whose clique-vertex incidence matrix is balanced.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 some graph classes which also lead to polynomial time or even linear time recognition algorithms within the corresponding subclasses.     

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.     

On minimally circular-imperfect graphs

A circular-perfect graph is a graph of which each induced subgraph has the same circular chromatic number as its circular clique number. In this paper, (1) we prove a lower bound on the order of minimally circular-imperfect graphs, and characterize those that attain the bound; (2) we prove that if G is a claw-free minimally circular-imperfect graph such that ω_c(G-x)>ω(G-x) for some x∈V(G), then G=K_(2k+1)/2+x for an integer k; and (3) we also characterize all minimally circular-imperfect line graphs.     

A Berge-keeping operation for graphs

We prove that a certain simple operation does not create odd holes or odd antiholes in a graph unless there are already some. In order to apply it, we need a vertex whose neighborhood has a coloring where the union of any two color classes is a connected graph; the operation is the shrinking of each of the color classes. Odd holes and antiholes do have such a vertex, and this property of minimal imperfect graphs implies the strong perfect graph theorem through the results of the paper. Conceivably, this property may be a target in the search for a proof of the strong perfect graph theorem different from the monumental achievement of Chudnovsky, Robertson, Seymour, and Thomas.     

Minimally non-Pfaffian graphs

We consider the question of characterizing Pfaffian graphs. We exhibit an infinite family of non-Pfaffian graphs minimal with respect to the matching minor relation. This is in sharp contrast with the bipartite case, as Little [C.H.C. Little, A characterization of convertible (0,1)-matrices, J. Combin. Theory Ser. B 18 (1975) 187–208] proved that every bipartite non-Pfaffian graph contains a matching minor isomorphic to K_3,3. We relax the notion of a matching minor and conjecture that there are only finitely many (perhaps as few as two) non-Pfaffian graphs minimal with respect to this notion.We define Pfaffian factor-critical graphs and study them in the second part of the paper. They seem to be of interest as the number of near perfect matchings in a Pfaffian factor-critical graph can be computed in polynomial time. We give a polynomial time recognition algorithm for this class of graphs and characterize non-Pfaffian factor-critical graphs in terms of forbidden central subgraphs.     

Circular chromatic number of triangle free graphs with given maximum average degree,extended abstract

In that paper we talk about triangle-free simple graphs with given maximum average degree less than 22/9. We give an upper bound for the circular chromatic number of such graphs which is at most 11/4. We give two other results whose proofs are omitted. We use a discharging method with forbidden configurations. We also omit in this paper the proof that they are indeed forbidden. These proofs rely on a recent paper [A. Raspaud, X. Zhu, List circular coloring of trees and cycles, (2006)] of Raspaud and Zhu and are all straight-forward.     

Two forbidden induced subgraphs and well-quasi-ordering

It is known that a class of graphs defined by a single forbidden induced subgraph G is well-quasi-ordered by the induced subgraph relation if and only if G is an induced subgraph of P₄. However, very little is known about well-quasi-ordered classes of graphs defined by more than one forbidden induced subgraph. We conjecture that for any natural number k, there are finitely many minimal classes of graphs defined by k forbidden induced subgraphs which are not well-quasi-ordered by the induced subgraph relation and prove the conjecture for k=2. We explicitly reveal many of the minimal classes defined by two forbidden induced subgraphs which are not well-quasi-ordered and many of those which are well-quasi-ordered by the induced subgraph relation.     

Triangle-free graphs and forbidden subgraphs

The relation of chromatic aspects and the existence of certain induced subgraphs of a triangle-free graph will be investigated. Based on a characterization statement of Pach, some results on the chromatic number of triangle-free graphs with certain forbidden induced subgraphs will be refined by describing their structure in terms of homomorphisms. In particular, we introduce chordal triangle-free graphs as a natural superclass of chordal bipartite graphs and describe the structure of the maximal triangle-free members. Finally, we improve on the upper bound for the chromatic number of triangle-free sK₂-free graphs by 1 for s2 giving the tight bound for s=3.