On self-duality and unigraphicity for $3$-polytopes

TL;DR

This paper characterizes degree sequences with a unique self-dual 3-polytope realization, extending existing algorithms with modifications to efficiently construct such polytopes.

Contribution

It provides a complete characterization of self-dual 3-polytopal degree sequences and introduces efficient algorithms for their construction.

Findings

Characterization of degree sequences with a unique self-dual 3-polytope realization

Development of linear-time algorithms for constructing these polytopes

Extensions and modifications of existing algorithms for better performance

Abstract

Recent literature posed the problem of characterising the graph degree sequences with exactly one $3$-polytopal (i.e. planar, $3$-connected) realisation. This seems to be a difficult problem in full generality. In this paper, we characterise the sequences with exactly one self-dual $3$-polytopal realisation. An algorithm in the literature constructs a self-dual $3$-polytope for any admissible degree sequence. To do so, it performs operations on the radial graph, so that the corresponding $3$-polytope and its dual are modified in exactly the same way. To settle our question and construct the relevant graphs, we apply this algorithm, we introduce some modifications of it, and we also devise new ones. The speed of these algorithms is linear in the graph order.