The non-commuting, non-generating graph of a non-simple group

Let G be a (finite or infinite) group such that G/Z(G) is not simple. The non-commuting, non-generating graph Ξ(G) of G has vertex set G\Z(G), with vertices x and y adjacent whenever [x, y] ̸= 1 and ⟨x, y⟩ ̸ = G. We investigate the relationship between the structure of G and the connectedness and diameter of Ξ(G). In particular, we prove that the graph either: (i) is connected with diameter at most 4; (ii) consists of isolated vertices and a connected component of diameter at most 4; or (iii) is the union of two connected components of diameter 2. We also describe in detail the finite groups with graphs of type (iii). In the companion paper [17], we consider the case where G/Z(G) is finite and simple.