New recursion formula for the interior polynomial based on non-expanding sets

TL;DR

This paper introduces a new, more transparent recursion formula for the interior polynomial of hypergraphs, based on non-expanding sets, providing clearer combinatorial and geometric insights.

Contribution

It presents a novel recursion formula for the interior polynomial that enhances understanding of its combinatorial and geometric properties.

Findings

New recursion formula based on non-expanding sets

Improved combinatorial interpretation of the interior polynomial

Enhanced connection to polyhedral geometry

Abstract

The interior polynomial was originally defined for hypergraphs and later shown to coincide with the Ehrhart polynomial of the root polytope of an associated bipartite graph. In previous work, we derived an alternating cycle recursion formula for the interior polynomial. Here, we introduce a new, more transparent recursion formula based on the structure of non-expanding sets. This formula offers a clearer combinatorial interpretation of the interior polynomial and its connection to polyhedral geometry.