Towards a comprehensive theory of conflict-tolerance graphs
Discrete Applied Mathematics
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In this article we introduce certain classes of graphs thatgeneralize φ-tolerance chain graphs. In a rank-tolerancerepresentation of a graph, each vertex is assigned two parameters:a rank, which represents the size of that vertex, and a tolerancewhich 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 withinvestigating the graph classes that arise from a variety offunctions, such as min, max, sum, and prod (product),that may be used as the coupling functions φ and ρ todefine the joint tolerance and the joint rank. Our goal is toobtain basic properties of the graph classes from basic propertiesof the coupling functions.We prove a skew symmetry result that when either φ or ρis continuous and weakly increasing, the(φ,ρ)-representable graphs equal the complements of the(ρ,φ)-representable graphs. In the case where either φor ρ is Archimedean or dual Archimedean, the class contains allthreshold graphs. We also show that, for min, max, sum, prod(product) and, in fact, for any piecewise polynomial φ, thereare infinitely many split graphs which fail to berepresentable.In the reflexive case (where φ = ρ), we show that ifφ is nondecreasing, weakly increasing and associative, theclass obtained is precisely the threshold graphs. This extends aresult of Jacobson, McMorris, and Mulder [10] for the functionmin to a much wider class, including max, sum, andprod.We also give results for homogeneous functions, powers of sums,and linear combinations of min and max. © 2006Wiley Periodicals, Inc. J Graph Theory