Separability and the genus of a partial dual

TL;DR

This paper explores how partial duality affects the genus of embedded graphs, linking it to graph separability, and characterizes partial duals in planar and projective embeddings.

Contribution

It establishes a connection between separability and genus in partial duals, providing characterizations for planar and projective embeddings.

Findings

Partial duality can change the genus of an embedded graph.

Separable structures determine the genus of partial duals.

A local move relates all partial duals in the plane and projective plane.

Abstract

Partial duality generalizes the fundamental concept of the geometric dual of an embedded graph. A partial dual is obtained by forming the geometric dual with respect to only a subset of edges. While geometric duality preserves the genus of an embedded graph, partial duality does not. Here we are interested in the problem of determining which edge sets of an embedded graph give rise to a partial dual of a given genus. This problem turns out to be intimately connected to the separability of the embedded graph. We determine how separability is related to the genus of a partial dual. We use this to characterize partial duals of graphs embedded in the plane, and in the real projective plane, in terms of a particular type of separation of an embedded graph. These characterizations are then used to determine a local move relating all partially dual graphs in the plane and in the real projective plane.