  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.
