Rank-tolerance graph classes

In this article we introduce certain classes of graphs that generalize φ-tolerance chain graphs. In a rank-tolerance representation of a graph, each vertex is assigned two parameters: a rank, which represents the size of that vertex, and a tolerance which represents an allowed extent of conflict with other vertices. Two vertices are adjacent if and only if their joint rank exceeds (or equals) their joint tolerance. This article is concerned with investigating the graph classes that arise from a variety of functions, such as min, max, sum, and prod (product), that may be used as the coupling functions φ and ρ to define the joint tolerance and the joint rank. Our goal is to obtain basic properties of the graph classes from basic properties of the coupling functions. We prove a skew symmetry result that when either φ or ρ is continuous and weakly increasing, the (φ, ρ)-representable graphs equal the complements of the (ρ, φ)-representable graphs. In the case where either φ or ρ is Archimedean or dual Archimedean, the class contains all threshold graphs. We also show that, for min, max, sum, prod (product) and, in fact, for any piecewise polynomial φ, there are infinitely many split graphs which fail to be representable. In the reflexive case (where φ=ρ), we show that if φ is nondecreasing, weakly increasing and associative, the class obtained is precisely the threshold graphs. This extends a result of Jacobson, McMorris, and Mulder [10] for the function min to a much wider class, including max, sum, and prod. We also give results for homogeneous functions, powers of sums, and linear combinations of min and max.