A characterization of robert's inequality for boxicity

AbstractF.S. Roberts defined the boxicity of a graph G as the smallest positive integer n for which there exists a function F assigning to each vertex x ϵG a sequence F(x)(1),F(x)(2),…, F(x)(n) of closed intervals of R so that distinct vertices x and y are adjacent in G if and only if F(x)(i)∩F(y)(i)≠∅ for i = 1, 2, 3, …, n. Roberts then proved that if G is a graph having 2n + 1 vertices, then the boxicity of G is at most n. In this paper, we provide an explicit characterization of this inequality by determining for each n ⩾ 1 the minimum collection Cn of graphs so that a graph G having 2n + 1 vertices has boxicity n if and only if it contains a graph from Cn as an induced subgraph. We also discuss combinatorial connections with analogous characterization problems for rectangle graphs, circular arc graphs, and partially ordered sets