Computing the boxicity of a graph by covering its complement by cointerval graphs

Abstract If F is a family of sets, its intersection graph has the sets in F as vertices and an edge between two sets if and only if they overlap. This paper investigates the concept of boxicity of a graph G, the smallest n such that G is the intersection graph of boxes in Euclidean n-space. The boxicity, b(G), was introduced by Roberts in 1969 and has since been studied by Cohen, Gabai, and Trotter. The concept has applications to niche overlap (competition) in ecology and to problems of fleet maintenance in operations research. These applications will be described briefly. While the problem of computing boxicity is in general a difficult problem (it is NP-complete), this paper develops techniques for computing boxicity which give useful bounds. They are based on the simple observation that b(G)≤k if and only if there is an edge covering of G by spanning subgraphs of G , each of which is a cointerval graph, the complement of an interval graph (a graph of boxicity ≤1.).