  An overlap representation is an assignment of sets to the vertices of a graph
in such a way that two vertices are adjacent if and only if the sets assigned
to them overlap. The overlap number of a graph is the minimum number of
elements needed to form such a representation. We find the overlap numbers of
cliques and complete bipartite graphs by relating the problem to previous
research in combinatorics. The overlap numbers of paths, cycles, and
caterpillars are also established. Finally, we show the NP-completeness of the
problems of extending an overlap representation and finding a minimum overlap
representation with limited containment.

Discrete Mathematics