On $3$-Colorable $P_5$-Free Graphs

Abstract

A graph is $P_5$-free when it does not contain a $P_5$ (that is, a path with five vertices) as an induced subgraph. The class of $P_5$-free graphs is of particular interest, especially with respect to the still unknown complexity status of the maximum stable set problem in that class. We investigate the class of $3$-colorable $P_5$-free graphs. We give a complete description of the structure of those graphs and derive a linear-time algorithm that tests membership in this class. Moreover, the algorithm is able to find a maximum weight stable set of a $3$-colorable $P_5$-free graph in linear time.