Abstract
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, W 4 , bull}-free, both 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
The graph G is coordinated if and only if every induced subgraph H of G belongs to Class 1
Summary
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. 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 [1,20]. In [14], clique-perfect graphs are characterized by minimal forbidden subgraphs for the class of chordal graphs. In [1] and [2], clique-perfect graphs are characterized by minimal forbidden subgraphs for two subclasses of clawfree graphs, and for Helly circular-arc graphs, respectively. 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
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