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 $t \geq 5$ remains little understood. 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.

中文翻译：

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在无 $P_t$ 的图中覆盖最小分隔符和潜在的最大派系

如果图不包含 $t$-vertex 路径作为诱导子图，则称为 $P_t$-free}。虽然无 $P_4$ 的图正是 cographs，但对于 $t \geq 5$ 的无 $P_t$ 的图的结构仍然知之甚少。一方面，对于任何固定的 $t$，经典的计算问题，例如最大权重独立集 (MWIS) 和 $3$-Coloring，在 $P_t$-free 图上并不知道是 NP-hard 问题。另一方面，尽管付出了巨大的努力，但最近才发现用于 MWIS 的多项式时间算法在 $P_6$-free 图中~[SODA 2019] 和 $3$-Coloring 在 $P_7$-free 图中~[Combinatorica 2018]。在这两种情况下，算法都依赖于对所考虑图类的深入结构洞察。$P_5$-free graphs~[SODA 2014] 和 $P_6$-free graphs~[SODA 2019] 中 MWIS 算法的主要工具之一是所谓的 Separator Covering Lemma，它断言在该图可以被恒定数量顶点的邻域的并集覆盖。在这篇笔记中，我们展示了这样的陈述可以推广到无 $P_7$ 的图，并且在无 $P_8$ 的图中是错误的。我们还讨论了这种用于覆盖具有邻域联合的潜在最大集团的声明的类似物。