On the Enumeration of Minimal Non-pairwise Compatibility Graphs

Abstract

A graph is a pairwise compatibility graph (PCG) if it can be represented by an edge weighted tree whose set of leaves is the set of vertices of the graph, and there is an edge between two vertices in the graph if and only if the distance between them in the tree is within a given interval. Enumerating all minimal non-PCGs (each of whose induced subgraphs is a PCG) with a given number of vertices is a challenging task, since it involves a large number of “configurations” that need to be inspected, an infinite search space of weights, and the construction of finite size evidence that a graph is not a PCG. We handle the problem of a large number of configurations by first screening graphs that are PCGs by using a heuristic PCG generator, and then constructing configurations that show some graphs to be PCGs. Finally, we generated configurations by excluding those configurations which cannot be used to show that a given graph is a PCG. To deal with the difficulty of infinite search space and construction of finite size evidence, we use linear programming (LP) formulations whose solutions serve as finite size evidence. We enumerated all minimal non-PCGs with nine vertices, the smallest integer for which minimal non-PCGs are unknown. We prove that there are exactly 1,494 minimal non-PCGs with nine vertices and provide evidence for each of them.