On Intersection Graphs of Spherical Caps

A graph G is called a cap graph if G represents the intersection graph of a family of spherical caps with the same angular radius. The complement of a graph G, denoted by $$\bar{G}$$ , is a graph with the same vertex set as G in which two vertices are adjacent only when they are non-adjacent in G. We prove that for any tree T, there is a real number $$\delta \in (0,\pi /2)$$ such that the complement of T is a cap graph of angular radius d for every $$d\in (\delta , \pi /2)$$ . We also classify all types of complete bipartite graphs that are cap graphs. For every such type we determine the range of angular radius of the spherical caps.