On unigraphic $3$-polytopes of radius one

TL;DR

This paper characterizes degree sequences that uniquely realize 3-polytopal graphs with radius one, providing exhaustive classifications for specific cases and computationally identifying all such polytopes up to certain sizes using advanced algorithms.

Contribution

It offers a complete classification of certain unigraphic 3-polytopal graphs and introduces a fast algorithm for enumerating these polytopes with radius one.

Findings

Identified all 3-polytopes of radius one with up to 17 vertices.

Classified degree sequences with specific properties that admit unique realizations.

Developed a high-performance algorithm for enumerating 3-polytopes.

Abstract

We ask which degree sequences admit a unique realisation as a $3$-polytopal graph (polyhedron) on $p$ vertices. We give an exhaustive list of these sequences for the case where one degree equals $p-1$ and exactly two or three of them equal $3$. We also find all $3$-polytopes of radius one with $p\leq 17$, and those with $q\leq 41$ edges, by developing a fast algorithm and making use of High Performance Computing.