Groups whose prime graph on class sizes has a cut vertex

Let G be a finite group, and let Δ(G) be the prime graph built on the set of conjugacy class sizes of G: this is the simple undirected graph whose vertices are the prime numbers dividing some conjugacy class size of G, two vertices p and q being adjacent if and only if pq divides some conjugacy class size of G. In the present paper, we classify the finite groups G for which Δ(G) has a cut vertex.