  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.
