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 for $t \geqslant 5$ remains little understood. On one hand, classic computational problems such Maximum Weight Independent Set (MWIS) and $3$-Coloring known to be NP-hard on any fixed $t$. other despite significant effort, polynomial-time algorithms MWIS in $P_6$-free graphs~[SODA 2019] $P_7$-free graphs~[Combinatorica 2018] have been found only recently. In both cases, rely deep structural insights into considered classes.
 One main tools $P_5$-free 2014] so-called Separator Covering Lemma that asserts every minimal separator can covered by union neighborhoods constant number vertices. this note we show statement generalizes false $P_8$-free graphs. We also discuss analogues covering potential maximal cliques with unions neighborhoods.