Classifying Families of Character Degree Graphs of Solvable Groups

We investigate prime character degree graphs of solvable groups. In particular, we consider a family of graphs ${\mathrm{\Gamma }}_{k,t}$ constructed by adjoining edges between two complete graphs in a one-to-one fashion. In this paper we determine completely which graphs ${\mathrm{\Gamma }}_{k,t}$ occur as the prime character degree graph of a solvable group.