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

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