Towards a comprehensive theory of conflict-tolerance graphs
Discrete Applied Mathematics
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Let φ be a symmetric binary function, positive valued onpositive arguments. A graph G = (V,E) is aφ-tolerance graph if each vertex υ ∈V can be assigned a closed intervalIυ and a positive tolerancetυ so that xy ∈ E↔ | Ix ∩ Iy|≥ φ(tx,ty). An Archimedeanfunction has the property of tending to infinity whenever one ofits arguments tends to infinity. Generalizing a known result of[15] for trees, we prove that every graph in a large class (whichincludes all chordless suns and cacti and the complete bipartitegraphs K2,k) is a φ-tolerance graphfor all Archimedean functions φ. This property does not holdfor most graphs. Next, we present the result that every graphG can be represented as a φG-tolerancegraph for some Archimedean polynomial φG.Finally, we prove that there is a "universal" Archimedean functionφ * such that every graph G is aφ*-tolerance graph. © 2002 Wiley Periodicals,Inc. J Graph Theory 41: 179194, 2002