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Types of embedded graphs and their Tutte polynomials

Part of: Graph theory

Published online by Cambridge University Press:  02 July 2019

STEPHEN HUGGETT  and
IAIN MOFFATT
STEPHEN HUGGETT
Affiliation:
Centre for Mathematical Sciences, University of Plymouth, Plymouth, PL4 8AA. e-mail: s.huggett@plymouth.ac.uk
IAIN MOFFATT
Affiliation:
Department of Mathematics, Royal Holloway, University of London, Egham, TW20 0EX. e-mail: iain.moffatt@rhul.ac.uk
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Abstract

We take an elementary and systematic approach to the problem of extending the Tutte polynomial to the setting of embedded graphs. Four notions of embedded graphs arise naturally when considering deletion and contraction operations on graphs on surfaces. We give a description of each class in terms of coloured ribbon graphs. We then identify a universal deletion-contraction invariant (i.e., a ‘Tutte polynomial’) for each class. We relate these to graph polynomials in the literature, including the Bollobás–Riordan, Krushkal and Las Vergnas polynomials, and give state-sum formulations, duality relations, deleton-contraction relations, and quasi-tree expansions for each of them.

MSC classification

Primary: 05C31: Graph polynomials
Secondary: 05C10: Planar graphs; geometric and topological aspects of graph theory
Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 169 , Issue 2 , September 2020 , pp. 255 - 297
Copyright
© Cambridge Philosophical Society 2019

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