Even Pairs in Claw-Free Perfect Graphs

An even pair in a graph is a pair of non-adjacent vertices such that every chordless path between them has even length. A graph is called strict quasi-parity when every induced subgraph that is not a clique has an even pair, and it is called perfectly contractile when every induced subgraph can be turned into a clique through a sequence of even-pair contractions. In this paper we determine theK1, 3-free graphs that are strict quasi-parity and those that are perfectly contractile. We show that for both classes the minimal forbidden configurations are odd holes, antiholes and some line-graphs of bipartite graphs, as conjectured by several authors. Our proofs are constructive and yield polynomial-time algorithms for the recognition of both classes.