Self-dual polyhedra of given degree sequence

TL;DR

This paper presents an algorithm for constructing self-dual polyhedra with specified degree sequences, analyzes minimal vertex counts for such polyhedra, and extends the approach to non-self-dual graphs with similar properties.

Contribution

It introduces a method to explicitly construct self-dual polyhedra based on degree sequences and determines minimal vertex counts for given degree constraints.

Findings

Algorithm for constructing self-dual polyhedra from degree sequences

Minimal number of vertices for self-dual polyhedra with given degrees

Construction of non-self-dual polyhedral graphs with specified degrees and face counts

Abstract

Given vertex valencies admissible for a self-dual polyhedral graph, we describe an algorithm to explicitly construct such a polyhedron. Inputting in the algorithm permutations of the degree sequence can give rise to non-isomorphic graphs. As an application, we find as a function of $n\geq 3$ the minimal number of vertices for a self-dual polyhedron with at least one vertex of degree $i$ for each $3\leq i\leq n$, and construct such polyhedra. Moreover, we find a construction for non-self-dual polyhedral graphs of minimal order with at least one vertex of degree $i$ and at least one $i$-gonal face for each $3\leq i\leq n$.