Elementary moves on lattice polytopes

TL;DR

This paper introduces a graph structure on Euclidean polytopes based on elementary vertex insertions and deletions, analyzing its connectivity and properties, especially within lattice polytopes.

Contribution

It defines a new graph framework on polytopes, explores its connectivity, and investigates special subgraphs related to lattice polytopes with fixed vertex counts.

Findings

The graph's connectivity properties are characterized.

Subgraphs induced by lattice polytopes with fixed vertices have intriguing features.

Several results on the structure and properties of these subgraphs are proved.

Abstract

We introduce a graph structure on Euclidean polytopes. The vertices of this graph are the $d$-dimensional polytopes contained in $\mathbb{R}^d$ and its edges connect any two polytopes that can be obtained from one another by either inserting or deleting a vertex, while keeping their vertex sets otherwise unaffected. We prove several results on the connectivity of this graph, and on a number of its subgraphs. We are especially interested in several families of subgraphs induced by lattice polytopes, such as the subgraphs induced by the lattice polytopes with $n$ or $n+1$ vertices, that turn out to exhibit intriguing properties.