On Local Operations that Preserve Symmetries and on Preserving Polyhedrality of Embeddings

TL;DR

This paper demonstrates that various local symmetry-preserving operations on embedded graphs maintain their polyhedral structure, generalizing previous results and connecting these operations to Delaney-Dress symbols.

Contribution

It provides a unified proof that multiple symmetry-preserving local operations preserve polyhedrality, extending Mohar's result for the dual operation and linking to Delaney-Dress symbols.

Findings

Symmetry-preserving operations maintain polyhedrality.

Connection established between local operations and Delaney-Dress symbols.

Generalization of polyhedrality preservation beyond dual operations.

Abstract

We prove that local operations that preserve all symmetries, as e.g. dual, truncation, ambo, or join,, as well as local operations that preserve all symmetries except orientation reversing ones, as e.g. gyro or snub, preserve the polyhedrality of simple embedded graphs. This generalizes a result by Mohar proving this for the operation dual. We give the proof based on an abstract characterization of these operations, prove that the operations are well defined, and also demonstrate the close connection between these operations and Delaney-Dress symbols. We also discuss more general operations not coming from 3-connected simple tilings of the plane.