The effect of symmetry‐preserving operations on 3‐connectivity

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.