Perfectly matchable set polynomials and h⁎-polynomials for stable set polytopes of complements of graphs

Abstract

A subset S of vertices of a graph G is called a perfectly matchable set of G if the subgraph induced by S contains a perfect matching. The perfectly matchable set polynomial of G, first made explicit by Ohsugi and Tsuchiya, is the (ordinary) generating function p(G;z) for the number of perfectly matchable sets of G.In this work, we provide explicit recurrences for computing p(G;z) for an arbitrary (simple) graph and use these to compute the Ehrhart h⁎-polynomials for certain lattice polytopes. Namely, we show that p(G;z) is the h⁎-polynomial for certain classes of stable set polytopes, whose vertices correspond to stable sets of G.