A sequence σ 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.
On Unigraphic Polyhedra with One Vertex of Degree p- 2
Maffucci R. W.
2024-01-01
Abstract
A sequence σ 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.
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