A method for enumerating pairwise compatibility graphs with a given number of vertices

Abstract

Azam et al. (2018) proposed a method to enumerate all pairwise compatibility graphs (PCGs) with a given number n of vertices. For a tuple (G,T,σ,λ) of a graph G with n vertices and a tree T with n leaves, a bijection σ between the vertices in G and the leaves in T, and a bi-partition λ of the set of non-adjacent vertex pairs in G, they formulated two linear programs, LP(G,T,σ,λ) and DLP(G,T,σ,λ) such that: exactly one of them is feasible; and G is a PCG if and only if LP(G,T,σ,λ) is feasible for some tuple (G,T,σ,λ). To reduce the number of graphs G with n vertices (resp., tuples) for which the LPs are solved, they excluded PCGs by heuristically generating PCGs (resp., some tuples that contain a sub-tuple (G′,T′,σ′,λ′) for n=4 whose LP(G′,T′,σ′,λ′) is infeasible). This paper proposes two improvements in the method: derive a sufficient condition for a graph to be a PCG for a given tree in order to exclude more PCGs; and characterize all sub-tuples (G′,T′,σ′,λ′) for n=4 for which LP(G′,T′,σ′,λ′) is infeasible, and enumerate tuples that contain no such sub-tuples by a branch-and-bound algorithm. Experimental results show that our method more efficiently enumerated all PCGs for n=8.