The class of graphs that are planar, 3-connected, of radius one, are exactly the 1-skeletons of polyhedra with one vertex adjacent to all others. Let F be a planar, 3-connected graph of radius one on p vertices, with a vertices of degree three. We characterise all unigraphic degree sequences for such graphs, when a ≥ 3 and p is large enough with respect to a. The proof methods reveal the structure of this class of graphs. We also solve the case a = 2 for any value of p.
Characterising 3-polytopes of radius one with unique realisation
Maffucci R. W.
2024-01-01
Abstract
The class of graphs that are planar, 3-connected, of radius one, are exactly the 1-skeletons of polyhedra with one vertex adjacent to all others. Let F be a planar, 3-connected graph of radius one on p vertices, with a vertices of degree three. We characterise all unigraphic degree sequences for such graphs, when a ≥ 3 and p is large enough with respect to a. The proof methods reveal the structure of this class of graphs. We also solve the case a = 2 for any value of p.
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