# Self-dual polyhedra of given degree sequence

## DOI:

https://doi.org/10.26493/2590-9770.1537.cf9## Keywords:

Polyhedron, Self-dual, Quadrangulation, Radial graph, Algorithm, Degree sequence## 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* ≥ 3 the minimal number of vertices for a self-dual polyhedron with at least one vertex of degree *i* for each 3 ≤ *i* ≤ *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 ≤ *i* ≤ *n*.

Another application is to rigidity theory, since the constructed families of polyhedra are generic circuits, and globally rigid in the plane.