  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.
