Four-vertex degree graphs of nonsolvable groups

For a finite group G, the character degree graph Δ(G) is the graph whose vertices are the primes dividing the degrees of the ordinary irreducible characters of G, with distinct primes p and q joined by an edge if pq divides some character degree of G. We determine all graphs with four vertices that occur as Δ(G) for some nonsolvable group G. Along with previously known results on character degree graphs of solvable groups, this completes the classification of all four-vertex graphs that occur as Δ(G) for some finite group G.