  Let $G$ be a finite group, and let $\Delta(G)$ be the prime graph built on
its set of conjugacy class sizes: this is the (simple undirected) graph whose
vertices are the prime numbers dividing some conjugacy class size of $G$, and
two distinct vertices $p, q$ are adjacent if and only if $pq$ divides some
class size of $G$. In this paper, we characterise the structure of those groups
$G$ whose prime graph $\Delta(G)$ is a block square.
