  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.
