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Canonical decompositions and algorithmic recognition of spatial graphs

Part of: Low-dimensional topology PL-topology

Published online by Cambridge University Press:  14 March 2024

Stefan Friedl Open the ORCID record for Stefan Friedl [Opens in a new window] ,
Lars Munser Open the ORCID record for Lars Munser [Opens in a new window] ,
José Pedro Quintanilha Open the ORCID record for José Pedro Quintanilha [Opens in a new window]  and
Yuri Santos Rego Open the ORCID record for Yuri Santos Rego [Opens in a new window]
Stefan Friedl
Affiliation:
Fakultät für Mathematik, Universität Regensburg, Regensburg, Germany (stefan.friedl@ur.de; lars.munser@ur.de)
Lars Munser
Affiliation:
Fakultät für Mathematik, Universität Regensburg, Regensburg, Germany (stefan.friedl@ur.de; lars.munser@ur.de)
José Pedro Quintanilha
Affiliation:
Fakultät für Mathematik, Universität Bielefeld, Bielefeld, Germany (jquintan@math.uni-bielefeld.de)
Yuri Santos Rego
Affiliation:
Fakultät für Mathematik (IAG), Otto-von-Guericke-Universität Magdeburg, Magdeburg, Germany (yuri.santos@ovgu.de)
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Abstract

We prove that there exists an algorithm for determining whether two piecewise-linear spatial graphs are isomorphic. In its most general form, our theorem applies to spatial graphs furnished with vertex colourings, edge colourings and/or edge orientations.

We first show that spatial graphs admit canonical decompositions into blocks, that is, spatial graphs that are non-split and have no cut vertices, in a suitable topological sense. Then, we apply a result of Haken and Matveev in order to algorithmically distinguish these blocks.

Keywords

spatial graphs3-manifolds with boundary patternHaken manifoldspiecewise-linear topology

MSC classification

Primary: 57M15: Relations with graph theory 57Q35: Embeddings and immersions
Secondary: 57Q40: Regular neighborhoods
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , First View , pp. 1 - 43
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on Behalf of The Edinburgh Mathematical Society

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