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

TL;DR

This paper introduces explicit recurrence relations for the perfectly matchable set polynomial of any graph and demonstrates its connection to the Ehrhart $h^*$-polynomials of stable set polytopes, linking combinatorics and polyhedral geometry.

Contribution

It provides a method to compute the perfectly matchable set polynomial for any graph and establishes its role as the $h^*$-polynomial for certain stable set polytopes.

Findings

Derived explicit recurrences for $p(G; z)$ for arbitrary graphs.

Connected $p(G; z)$ to Ehrhart $h^*$-polynomials of stable set polytopes.

Identified classes of stable set polytopes with $h^*$-polynomials equal to $p(G; z)$.

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 $G$.