Self-Reachable Configuration Polytopes for Trees

TL;DR

This paper investigates special lattice polytopes derived from self-reachable chip configurations on trees, proving their integer decomposition property, characterizing their vertices, and showing equivalence to a unit cube in minimal cases.

Contribution

It introduces and characterizes self-reachable configuration polytopes on trees, establishing their properties and equivalences, which is a novel contribution to combinatorial geometry.

Findings

Polytopes always have the integer decomposition property.

Vertex sets of these polytopes are characterized.

Minimal chip configurations lead to polytopes equivalent to a unit cube.

Abstract

We study lattice polytopes which arise as the convex hull of chip vectors for \textit{self-reachable} chip configurations on a tree $T$. We show that these polytopes always have the integer decomposition property and characterize the vertex sets of these polytopes. Additionally, in the case of self-reachable configurations with the smallest possible number of chips, we show that these polytopes are unimodularly equivalent to a unit cube.