Critical classes of power graphs and reconstruction of directed power graphs

TL;DR

This paper introduces new concepts related to power graphs of finite groups, correcting previous errors and providing an algorithm to reconstruct directed power graphs from undirected ones.

Contribution

It defines critical classes and Moore closure in power graphs, correcting prior work and offering a reconstruction algorithm for directed power graphs.

Findings

Corrected a mistake in a key theorem about power graphs.

Developed an algorithm to reconstruct directed power graphs from undirected ones.

Explored properties of closed twin classes in power graphs.

Abstract

In a graph $\Gamma=(V,E)$, we consider the common closed neighbourhood of a subset of vertices and use this notion to introduce a Moore closure operator in $V.$ We also consider the closed twin equivalence relation in which two vertices are equivalent if 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 classes. The study of critical classes, together with properties of the Moore closure operator, allow us to correct a mistake in the proof of {\rm \cite[Theorem 2 ]{Cameron_2}} and to deduce a simple algorithm to reconstruct the directed power graph of a finite group from its undirected counterpart, as asked in \cite[Question 2]{GraphsOnGroups}.