The overlap number of a graph

Details The overlap number of a graph The overlap number of a graph

2010-08-12

arxiv.org/abs/1008.2170v2

Computer Science

Discrete Mathematics

20 pages, 2 figures

Scientific paper

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.

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