Forbidden subgraphs of power graphs

TL;DR

This paper investigates the structural properties of power graphs of groups, identifying which groups produce graphs belonging to specific classes like perfect, threshold, or split graphs, with complete results for nilpotent groups.

Contribution

It provides a comprehensive analysis of forbidden subgraphs in power graphs, offering complete characterizations for nilpotent groups and partial results for broader classes.

Findings

Power graphs of groups are always perfect.

Groups with threshold or split power graphs are fully characterized.

Open problems remain for non-nilpotent groups.

Abstract

The undirected power graph (or simply power graph) of a group $G$, denoted by $P(G)$, is a graph whose vertices are the elements of the group $G$, in which two vertices $u$ and $v$ are connected by an edge between if and only if either $u=v^i$ or $v=u^j$ for some $i$, $j$. A number of important graph classes, including perfect graphs, cographs, chordal graphs, split graphs, and threshold graphs, can be defined either structurally or in terms of forbidden induced subgraphs. We examine each of these five classes and attempt to determine for which groups $G$ the power graph $P(G)$ lies in the class under consideration. We give complete results in the case of nilpotent groups, and partial results in greater generality. In particular, the power graph is always perfect; and we determine completely the groups whose power graph is a threshold or split graph (the answer is the same for both classes). We give a number of open problems.