Perfectly Matchable Set Polynomials and $h^*$-polynomials for Stable Set Polytopes of Complements of Graphs

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 $G$.