On unigraphic polyhedra with one vertex of degree $p-2$

TL;DR

This paper characterizes and classifies degree sequences of polyhedral graphs that are uniquely realizable as polyhedra, focusing on sequences with a maximum degree of p-2 and exploring their structural properties.

Contribution

It provides a classification of unigraphic polyhedral degree sequences with specific degree constraints, advancing understanding of unique polyhedral realizations.

Findings

Classified sequences starting with p-2,p-2

Characterized sequences starting with p-2 containing one 3

Identified sequences with a few high-degree vertices

Abstract

A sequence $\sigma$ of $p$ non-negative integers is unigraphic if it is the degree sequence of exactly one graph, up to isomorphism. A polyhedral graph is a $3$-connected, planar graph. We investigate which sequences are unigraphic with respect to the class of polyhedral graphs, meaning that they admit exactly one realisation as a polyhedron. We focus on the case of sequences with largest entry $p-2$. We give a classification of polyhedral unigraphic sequences starting with $p-2,p-2$, as well as those starting with $p-2$ and containing exactly one $3$. Moreover, we characterise the unigraphic sequences where a few vertices are of high degree. We conclude with a few other examples of families of unigraphic polyhedra.