Forbidden Subgraphs in Intersection Power Graphs of Finite Groups

The intersection power graph of a finite group G is a simple graph whose vertex set is G, in which two distinct vertices x and y are adjacent if and only if either one of x and y is the identity element, or < x >boolean AND < y > is non-trivial. A number of important graph classes, including cographs, chordal graphs, split graphs, and threshold graphs, can be defined either structurally or in terms of forbidden induced subgraphs. In this paper, we characterize the finite groups whose intersection power graphs are cographs, split graphs, and threshold graphs. We also classify the finite nilpotent groups whose intersection power graphs are chordal.