ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 24 (2024) #P2.01
https://doi.org/10.26493/1855-3974.2749.b64
(Also available at http://amc-journal.eu)
On local operations that preserve symmetries
and on preserving polyhedrality of maps
Gunnar Brinkmann , Heidi Van den Camp *
Ghent University, Krijgslaan 281 S9, Ghent, Belgium
Received 1 December 2021, accepted 8 July 2023, published online 4 September 2023
Abstract
We prove that local operations that preserve all symmetries, as e.g. dual, truncation, me-
dial, or join, as well as local operations that are only guaranteed to preserve all orientation-
preserving symmetries, as e.g. gyro or snub, preserve the polyhedrality of simple maps.
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.
Keywords: Embedded graph, map, polyhedral embedding, operation, symmetry, tiling.
Math. Subj. Class. (2020): 05C76, 05C10, 05C40, 52B05
*Corresponding author.
E-mail addresses: Gunnar.Brinkmann@ugent.be (Gunnar Brinkmann), heidi.vandencamp@gmail.com
(Heidi Van den Camp)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
ISSN 1855-3966 (tiskana izd.), ISSN 1855-3974 (elektronska izd.)
ARS MATHEMATICA CONTEMPORANEA 24 (2024) #P2.01
https://doi.org/10.26493/1855-3974.2749.b64
(Dostopno tudi na http://amc-journal.eu)
Lokalne operacije, ki ohranjajo simetrije,
in ohranjanje poliedrskosti zemljevidov
Gunnar Brinkmann , Heidi Van den Camp *
Ghent University, Krijgslaan 281 S9, Ghent, Belgium
Prejeto 1. decembra 2021, sprejeto 8. julija 2023, objavljeno na spletu 4. septembra 2023
Povzetek
Dokažemo, da lokalne operacije, ki ohranjajo vse simetrije, kot so npr. dual, obrezo-
vanje, medial in spoj, kot tudi lokalne operacije, za katere je zagotovljeno le, da ohranjajo
vse simetrije, ki ohranjajo orientacijo, kot sta npr. gyro ali snub, ohranjajo poliedrskost
enostavnih zemljevidov. To je posplošitev Moharjevega rezultata; on je to dokazal za ope-
racijo dual. Naš dokaz temelji na abstraktni karakterizaciji teh operacij; dokažemo, da
so te operacije dobro definirane, prikažemo pa tudi tesno povezavo med njimi in simboli
Delaneya in Dressa.
Ključne besede: Vloženi graf, zemljevid, poliedrska vložitev, operacija, simetrija, tlakovanje.
Math. Subj. Class. (2020): 05C76, 05C10, 05C40, 52B05
*Kontaktni avtor.
E-poštna naslova: Gunnar.Brinkmann@ugent.be (Gunnar Brinkmann), heidi.vandencamp@gmail.com
(Heidi Van den Camp)
cb To delo je objavljeno pod licenco https://creativecommons.org/licenses/by/4.0/