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

Flavia Bonomo,Guillermo Durán,Gabriel Sueiro,Francisco Soulignac

doi:10.1016/j.endm.2008.01.010

Abstract

Abstract 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,W 4 ,bull}-free, two superclasses of triangle-free graphs.

Highlights

Let G be a simple finite undirected graph, with vertex set V (G) and edge set E(G)
Given two graphs G and G, we say that G contains G if G is isomorphic to an induced subgraph of G
A graph G is clique-perfect if τC (H) = αC (H) for every induced subgraph H of G

Summary

Introduction

Let G be a simple finite undirected graph, with vertex set V (G) and edge set E(G). Denote by G the complement of G. A graph G is clique-perfect if τC (H) = αC (H) for every induced subgraph H of G. Finding the complete lists of minimal forbidden induced subgraphs for the classes of clique-perfect and coordinated graphs turns out to be a difficult task [2,24]. In [16], clique-perfect graphs are characterized by minimal forbidden subgraphs for the class of chordal graphs. We characterize clique-perfect and coordinated graphs by minimal forbidden induced subgraphs when the graph lies in one of two superclasses of triangle-free graphs: paw-free and {gem, W4, bull}-free graphs. We prove that in these cases both classes are equivalent to perfect graphs and, in consequence, the only forbidden subgraphs are the odd holes (odd antiholes of length at least seven are neither paw-free nor {gem, W4, bull}-free). We can deduce polynomial-time algorithms to recognize clique-perfect and coordinated graphs when the graph belongs to these classes

Superclasses of triangle-free graphs

Paw-free graphs

Summary