A graph is
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Abstract strongly perfect if every induced subgraph of it has a stable set that meets every maximal clique of . A graph isclaw‐free if no vertex has three pairwise nonadjacent neighbors. The characterization of claw‐free graphs that are strongly perfect by a set of forbidden induced subgraphs was conjectured by Ravindra in 1990 and was proved by Wang in 2006. Here we give a shorter proof of this characterization. -
null (Ed.)An odd hole in a graph is an induced cycle with odd length greater than 3. In an earlier paper (with Sophie Spirkl), solving a longstanding open problem, we gave a polynomial-time algorithm to test if a graph has an odd hole. We subsequently showed that, for every t , there is a polynomial-time algorithm to test whether a graph contains an odd hole of length at least t . In this article, we give an algorithm that finds a shortest odd hole, if one exists.more » « less
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null (Ed.)The claw is the graph $K_{1,3}$, and the fork is the graph obtained from the claw $K_{1,3}$ by subdividing one of its edges once. In this paper, we prove a structure theorem for the class of (claw, $C_4$)-free graphs that are not quasi-line graphs, and a structure theorem for the class of (fork, $C_4$)-free graphs that uses the class of (claw, $C_4$)-free graphs as a basic class. Finally, we show that every (fork, $C_4$)-free graph $G$ satisfies $\chi(G)\leqslant \lceil\frac{3\omega(G)}{2}\rceil$ via these structure theorems with some additional work on coloring basic classes.more » « less
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null (Ed.)Let C be a class of graphs closed under taking induced subgraphs. We say that C has the clique-stable set separation property if there exists c ∈ N such that for every graph G ∈ C there is a collection P of partitions (X, Y ) of the vertex set of G with |P| ≤ |V (G)| c and with the following property: if K is a clique of G, and S is a stable set of G, and K ∩ S = ∅, then there is (X, Y ) ∈ P with K ⊆ X and S ⊆ Y . In 1991 M. Yannakakis conjectured that the class of all graphs has the clique-stable set separation property, but this conjecture was disproved by M. G¨o¨os in 2014. Therefore it is now of interest to understand for which classes of graphs such a constant c exists. In this paper we define two infinite families S, K of graphs and show that for every S ∈ S and K ∈ K, the class of graphs with no induced subgraph isomorphic to S or K has the clique-stable set separation property.more » « less
