Covering Minimal Separators And Potential Maximal Cliques In P-T-Free Graphs

A graph is called P-t-free if it does not contain a t-vertex path as an induced subgraph. While P-4-free graphs are exactly cographs, the structure of P-t-free graphs for t >= 5 remains not well-undestood. On one hand, classic computational problems such as MAXIMUM WEIGHT INDEPENDENT SET (MWIS) and 3-COLORING are not known to be NP-hard on P-t-free graphs for any fixed t. On the other hand, despite significant effort, polynomial-time algorithms for MWIS in P-6-free graphs [SODA 2019] and 3-COLORING in P-7-free graphs [Combinatorica 2018] have been found only recently. In both cases, the algorithms rely on deep structural insights into the considered graph classes.One of the main tools in the algorithms for MWIS in P-5-free graphs [SODA 2014] and in P-6-free graphs [SODA 2019] is the so-called Separator Covering Lemma that asserts that every minimal separator in the graph can be covered by the union of neighborhoods of a constant number of vertices. In this note we show that such a statement generalizes to P-7-free graphs and is false in P-8-free graphs. We also discuss analogues of such a statement for covering potential maximal cliques with unions of neighborhoods.