On mirror nodes in graphs without long induced paths

The notion of mirror nodes, first introduced by Fomin, Grandoni and Kratsch in 2006, turned out to be a useful item for their design of an algorithm for the maximum independent set problem (MIS). Given two nodes u, v of a graph G = (V, E) with distG(u, v) = 2, u is called a mirror of v if N(v) \N(u) induces a (possibly empty) clique. In order to have a well defined term, we add that every complete graph has a mirror node. In general there might be no mirror nodes in arbitrary graphs, e.g. there exist graphs with no induced paths of length six without any mirror node. But in contrast every P4-free graph contains a mirror node. Therefore we are interested in the existence of mirror nodes in P5-free graphs. This class is well-studied in the context of MIS. For several subfamilies of P5-free graphs with an additional forbidden induced subgraph, we demonstrate that mirror nodes always occur. Whether there always exists a mirror node in P5-free graphs is still open, even in (P5, C5)-free and (P5, P 5)-free graphs.