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

Let G be a finite group. Denoting by 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 cd(G), and two distinct vertices p, q are adjacent if and only if pq divides some number in 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.