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Let $G$ be a finite group, and let $\Delta(G)$ be the prime graph built onthe set of conjugacy class sizes of $G$: this is the simple undirected graphwhose 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 someconjugacy class size of $G$. In the present paper, we classify the finitegroups $G$ for which $\Delta(G)$ has a cut vertex.

Groups whose prime graph on class sizes has a cut vertex