Characterising $3$-polytopes of radius one with unique realisation

Abstract

Let $F$ be a planar, $3$-connected graph of radius one on $p$ vertices, with $a$ vertices of degree three. We characterise all unigraphic degree sequences for such graphs, when $a\geq 3$ and $p$ is large enough with respect to $a$. This complements the work of \cite{mafpo4} for $a\leq 3$.