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arXiv · 2301.08021

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

Jim Delitroz·Riccardo W. Maffucci
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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.

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Jim Delitroz, Riccardo W. Maffucci. 2023-01-19. On unigraphic polyhedra with one vertex of degree $p-2$. https://arxiv.org/abs/2301.08021

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