Self-dual polyhedra of given 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.