  In a graph $\Gamma=(V,E)$, we consider the closed neighbourhood of a subset
of vertices and use this notion to deduce a Moore closure operator in $V.$ We
also consider the closed twin equivalence relation which puts in relation two
vertices when they have the same closed neighbourhood. Those notions are deeply
explored when $\Gamma$ is the power graph associated with a finite group $G$.
In that case, among the corresponding closed twin equivalence classes, we
introduce the concepts of plain, compound and critical class. The study of
critical classes, joined with the properties of the Moore closure operator,
allows to correct a mistake in the proof of Theorem 2 in P. J. Cameron, The
power graph of a finite group, II, Journal of Group Theory 13 (2010) and to
deduce a simple algorithm to reconstruct the directed power graph of a finite
group from its undirected counterpart, as asked in Question 2 in P. J. Cameron,
Graphs defined on groups, International Journal of Group Theory, vol. 11
(2022).
