All variations on perfectly orderable graphs

An ordered graph is a graph whose vertices are positive integers. Two ordered graphs are isomorphic if the order-preserving bijection between their sets of vertices is a graph isomorphism. We identify the family of all sets S of ordered graphs with the following properties: (1) Each member of S is a P4 (defined as a chordless path with four vertices and three edges). (2) If an ordered graph Z has no induced subgraph isomorphic (as an ordered graph) to a member of S, then Z is perfect. This work is related to Berge's Strong Perfect Graph Conjecture and was motivated by Chvatal's theorem on perfectly orderable graphs.