The Connectivity of the Dual

TL;DR

This paper investigates the conditions under which the dual of a highly connected embedded graph on surfaces of higher genus remains highly connected, providing bounds and exact values for specific cases like the torus.

Contribution

It establishes that no universal connectivity guarantees the dual's connectivity across all genera and determines exact connectivity thresholds for the torus.

Findings

No universal connectivity guarantees dual connectivity for all genera.

Six-connected graphs on the torus have 3-connected duals.

Four-connected graphs on the torus have 2-connected duals.

Abstract

The 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 this and related questions. Among other things, we prove that there is no connectivity that for every genus guarantees the 3-connectedness or 2-connectedness of the dual, and give upper bounds for the minimum genus for which (with c>2) a c-connected embedded graphs with a dual that has a 1- or 2-cut can occur. For the torus, we determine exact values for the connectivity needed to guarantee 3- respectively 2-connectivity of the dual. 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 non-complete graphs.