Forbidden subgraphs of power graphs

Pallabi Manna, Peter J. Cameron, Ranjit Mehatari*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

13 Citations (Scopus)

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 = vi or v = uj 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.
Original languageEnglish
Article numberP3.4
Number of pages14
JournalElectronic Journal of Combinatorics
Volume28
Issue number3
DOIs
Publication statusPublished - 2 Jul 2021

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