Basic perfect graphs and their extensions

Abstract

In this article, we present a characterization of basic graphs in terms of forbidden induced subgraphs. This class of graphs was introduced by Conforti et al. (Square-free perfect graphs, J. Combin. Theory Ser. B, 90 (2) (2004) 257–307), and it plays an essential role in the announced proof of the Strong Perfect Graph Conjecture by Chudnovsky et al. ( http://arxiv.org/PS_cache/math/pdf/0212/0212070.pdf). Let G and H be graphs. A substitution of H in G replacing a vertex v ∈ V ( G ) is the graph G ( v → H ) consisting of disjoint union of H and G - v with the additional edge-set { xy : x ∈ V ( H ) , y ∈ N G ( v ) } . For a class of graphs P , its substitutional closure P * consists of all graphs that can be obtained from graphs of P by repeated substitutions. We apply the reducing pseudopath method (Discrete Appl. Math. 128 (2–3) (2003) 487–509) to characterize the substitutional closure of the class of basic graphs in terms of forbidden induced subgraphs.