On Unigraphic Polyhedra with One Vertexof Degreep-2

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.