Partial duality and closed 2-cell embeddings

TL;DR

This paper explores the concept of partial duality in graph embeddings, providing new descriptions and criteria for when a partial dual results in a closed 2-cell embedding, which is important for topological graph theory.

Contribution

It introduces alternative descriptions of partial duality and establishes necessary and sufficient conditions for partial duals to be closed 2-cell embeddings.

Findings

Derived a necessary and sufficient condition for partial duals to be closed 2-cell

Provided a sufficient condition for no partial dual to be closed 2-cell

Demonstrated symmetry between vertices and faces in partial duality

Abstract

In 2009 Chmutov introduced the idea of partial duality for embeddings of graphs in surfaces. We discuss some alternative descriptions of partial duality, which demonstrate the symmetry between vertices and faces. One is in terms of band decompositions, and the other is in terms of the gem (graph-encoded map) representation of an embedding. We then use these to investigate when a partial dual is a closed 2-cell embedding, in which every face is bounded by a cycle in the graph. We obtain a necessary and sufficient condition for a partial dual to be closed 2-cell, and also a sufficient condition for no partial dual to be closed 2-cell.