On unigraphic $3$-polytopes of radius one

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 $p\le 17$, and those with $q\le 41$ edges, by developing a fast algorithm and making use of High Performance Computing.