The connectivity of the dual

Abstract

AbstractThe dual of a polyhedron is a polyhedron—or in graph‐theoretical terms: the dual of a 3‐connected plane graph is a 3‐connected plane graph. Astonishingly, except for sufficiently large facewidth, not much is known about the connectivity of the dual on higher surfaces. Are the duals of 3‐connected embedded graphs of higher genus 3‐connected, too? If not, which connectivity guarantees 3‐connectedness of the dual? In this article, we give answers to some of these and related questions. We prove that there is no connectivity that guarantees the 3‐connectedness or 2‐connectedness of the dual for every genus, and give upper bounds for the minimum genus for which (with ) a c‐connected embedded graph with a dual that has a 1‐ or 2‐cut can occur. We prove that already on the torus, we need 6‐connectedness to guarantee 3‐connectedness of the dual and 4‐connectedness to guarantee 2‐connectedness of the dual. In the last section, we answer a related question by Plummer and Zha on orientable embeddings of highly connected noncomplete graphs.