Non-solvable Groups whose Character Degree Graph has a Cut-Vertex. I

Abstract Let G be a finite group. Denoting by $$\textrm{cd}(G)$$ cd ( G ) the set of degrees irreducible complex characters , we consider character degree graph : this is (simple undirected) whose vertices are prime divisors numbers in and two distinct p q adjacent if only pq divides some number . In series three papers starting with present one, analyze structure non-solvable groups possesses cut-vertex i.e. vertex removal increases connected components graph.