Forbidden subgraphs in generating graphs of finite groups

Let G be a 2-generated finite group. The generating graph Γ(G) is the graph whose vertices are the elements of G and where two vertices g1 and g2 are adjacent if G = 〈g1, g2〉. This graph encodes the combinatorial structure of the distribution of generating pairs across G. In this paper we study some graph theoretic properties of Γ(G), with particular emphasis on those properties that can be formulated in terms of forbidden induced subgraphs. In particular we investigate when the generating graph Γ(G) is a cograph (giving a complete description when G is soluble) and when it is perfect (giving a complete description when G is nilpotent and proving, among other things, that Γ(Sn) and Γ(An) are perfect if and only if n ≼ 4). Finally we prove that for a finite group G, the properties that Γ(G) is split, chordal or C4-free are equivalent.