Solvable groups whose prime divisor character degree graphs are 1-connected

A cut vertex of a graph is a vertex whose removal causes the resulting graph to have more connected components. We show that the prime divisor character degree graph of a solvable group has at most one cut vertex. We also prove that a solvable group whose prime divisor character degree graph has a cut vertex has at most two normal nonabelian Sylow subgroups. We determine the structures of those solvable groups whose prime divisor character degree graph has a cut vertex and has two normal nonabelian Sylow subgroups. Finally, we characterize a subgroup determined by the prime associated with the cut vertex in terms of the structure of the prime divisor character degree graph.