Stable sets in k-colorable [formula omitted]-free graphs

Abstract

We show that, for fixed k, there is a polynomial-time algorithm that finds a maximum (or maximum-weight) stable set in any graph that belongs to the class of k-colorable P 5 -free graphs, or, more generally, to the class of P 5 -free graphs that contain no clique of size k + 1 . This is based on the following structural result: every connected k-colorable P 5 -free graph has a vertex whose non-neighbors induce a ( k − 1 ) -colorable subgraph.