The Effect of Symmetry-preserving Operations on 3-Connectivity

TL;DR

This paper characterizes which symmetry-preserving operations on embedded graphs maintain 3-connectivity, extending previous results to graphs with face-width less than three.

Contribution

It provides a simple condition that exactly characterizes lopsp-operations preserving 3-connectivity for all embedded graphs.

Findings

Identifies conditions under which lopsp-operations preserve 3-connectivity.

Extends preservation results to graphs with face-width less than three.

Offers a unified framework for understanding symmetry-preserving graph operations.

Abstract

In 2017, Brinkmann, Goetschalckx and Schein introduced a very general way of describing operations on embedded graphs that preserve all orientation-preserving symmetries of the graph. This description includes all well-known operations such as Dual, Truncation and Ambo. As these operations are applied locally, they are called local orientation-preserving symmetry-preserving operations (lopsp-operations). In this text we will use the general description of these operations to determine their effect on 3-connectivity. Recently it was proved that all lopsp-operations preserve 3-connectivity of graphs that have face-width at least three. We present a simple condition that characterises exactly which lopsp-operations preserve 3-connectivity for all embedded graphs, even for those with face-width less than three.