Character degree graphs that are complete graphs

Throughout this note G will be a finite group and cd(G) will be the set of degrees of irreducible characters of G. A theorem of Thompson says that if every degree in cd(G) − {1} is divisible by some prime p, then G has a normal p-complement (see Corollary 12.2 of [5] or Theorem 23.3 of [4]). In [2], Berkovich showed that more can be said in this situation. In particular, he prove