On Unigraphic Polyhedra with One Vertex of Degree \({p-2}\)

A sequence \(\sigma \) $\sigma$ 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 \(p-2\) $p-2$ . We give a classification of polyhedral unigraphic sequences starting with \(p-2,p-2\) $p-2,p-2$ , as well as those starting with \(p-2\) $p-2$ 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.