A new characterization of [formula omitted]-free graphs

Abstract

We study P 6 -free graphs, i.e., graphs that do not contain an induced path on six vertices. Our main result is a new characterization of this graph class: a graph G is P 6 -free if and only if each connected induced subgraph of G on more than one vertex contains a dominating induced cycle on six vertices or a dominating (not necessarily induced) complete bipartite subgraph. This characterization is minimal in the sense that there exists an infinite family of P 6 -free graphs for which a smallest connected dominating subgraph is a (not induced) complete bipartite graph. Our characterization of P 6 -free graphs strengthens results of Liu and Zhou, and of Liu, Peng and Zhao. Our proof has the extra advantage of being constructive: we present an algorithm that finds such a dominating subgraph of a connected P 6 -free graph in polynomial time. This enables us to solve the Hypergraph 2- Colorability problem in polynomial time for the class of hypergraphs with P 6 -free incidence graphs.