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

2021/10/12 by Gunnar Brinkmann, Brinkmann, Gunnar, Heidi Van den Camp +1
Computer Science · Mathematics · #Advanced Combinatorial Mathematics #Advanced Graph Theory Research #Cellular Automata and Applications #Combinatorics (math.CO) #FOS: Mathematics

paper · pdf · doi:10.48550/arxiv.2110.06047

openalex publication_date 2021/10/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

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