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Complexity of the Bollobás–Riordan Polynomial. Exceptional Points and Uniform Reductions

Markus Bläser, Holger Dell, Johann A. Makowsky
2009 Theory of Computing Systems  
The coloured Tutte polynomial by Bollobás and Riordan is, as a generalization of the Tutte polynomial, the most general graph polynomial for coloured graphs that satisfies certain contraction-deletion  ...  complexity of graph polynomials.  ...  We classify the complexity of evaluating the Bollobás-Riordan polynomial in the following way. Main Theorem. Let σ be an evaluation point.  ... 
doi:10.1007/s00224-009-9213-7 fatcat:7uafgxlslffwdeswh7di6nsmjy

Complexity of the Bollobás-Riordan Polynomial [chapter]

Markus Bläser, Holger Dell, Johann A. Makowsky
Computer Science – Theory and Applications  
The coloured Tutte polynomial by Bollobás and Riordan is, as a generalization of the Tutte polynomial, the most general graph polynomial for coloured graphs that satisfies certain contraction-deletion  ...  complexity of graph polynomials.  ...  We classify the complexity of evaluating the Bollobás-Riordan polynomial in the following way. Main Theorem. Let σ be an evaluation point.  ... 
doi:10.1007/978-3-540-79709-8_12 dblp:conf/csr/BlaserDM08 fatcat:dzxainmgo5cjpgth3s3ww6kkja

Quasi-tree expansion for the Bollobás-Riordan-Tutte polynomial

Abhijit Champanerkar, Ilya Kofman, Neal Stoltzfus
2011 Bulletin of the London Mathematical Society  
We generalize the spanning tree expansion of the Tutte polynomial to a quasi-tree expansion of the Bollob\'as-Riordan-Tutte polynomial.  ...  The Bollob\'as-Riordan-Tutte polynomial is a three-variable polynomial that extends the Tutte polynomial to oriented ribbon graphs.  ...  Summary of results Bollobás and Riordan extended the Tutte polynomial to an invariant of oriented ribbon graphs, now called the Bollobás-Riordan-Tutte (BRT) polynomial.  ... 
doi:10.1112/blms/bdr034 fatcat:7acylrioonbanh5yqrmocwapuy

Generalization of the Bollobás-Riordan polynomial for tensor graphs

Adrian Tanasa
2011 Journal of Mathematical Physics  
This polynomial is a natural generalization of the Bollobás-Riordan polynomial (used to characterize matrix graphs) and is different of the Gur polynomial, (R.  ...  The polynomial T is defined for both colorable and non-colorable graphs and it is proved to satisfy the contraction/deletion relation. A non-trivial example of a non-colorable graphs is analyzed.  ...  Acknowledgments The author acknowledges the grant PN 09 37 01 02 and the CNCSIS grants "Tinere echipe" 77/04.08.2010 and "Idei" 454/2009, ID-44.  ... 
doi:10.1063/1.3605312 fatcat:koqwkc5j6fap3cnyvdxo73yhfq

Topological Tutte Polynomial [article]

Sergei Chmutov
2017 arXiv   pre-print
This is a survey recent works on topological extensions of the Tutte polynomial.  ...  The arrow version of the Bollobás-Riordan polynomial.  ...  The dichromatic version of the Bollobás-Riordan polynomial.  ... 
arXiv:1708.08132v1 fatcat:v22y3mf42nfpjfus3k7sgtxdr4

The Jones polynomial and graphs on surfaces

Oliver T. Dasbach, David Futer, Efstratia Kalfagianni, Xiao-Song Lin, Neal W. Stoltzfus
2008 Journal of combinatorial theory. Series B (Print)  
The Bollobas-Riordan-Tutte polynomial generalizes the Tutte polynomial of planar graphs to graphs that are embedded in closed oriented surfaces of higher genus.  ...  In this paper we show that the Jones polynomial of any link can be obtained from the Bollobas-Riordan-Tutte polynomial of a certain oriented ribbon graph associated to a link projection.  ...  Acknowledgments We would like to thank Sergei Chmutov and Igor Pak for helpful discussions on the Bollobás-Riordan-Tutte polynomial.  ... 
doi:10.1016/j.jctb.2007.08.003 fatcat:fwxsx42do5fvfehaqzzebeb3km

The chromatic polynomial of fatgraphs and its categorification

Martin Loebl, Iain Moffatt
2008 Advances in Mathematics  
We extend our construction and categorify the Bollobas-Riordan polynomial (a generalisation of the Tutte polynomial to embedded graphs).  ...  Motivated by Khovanov homology and relations between the Jones polynomial and graph polynomials, we construct a homology theory for embedded graphs from which the chromatic polynomial can be recovered  ...  M.L. gratefully acknowledges the support of CONICYT via grant Anillo en Redes.  ... 
doi:10.1016/j.aim.2007.11.016 fatcat:dkwh42hzdrh2zhrxl4reficc7i

Generalized duality for graphs on surfaces and the signed Bollobas-Riordan polynomial [article]

Sergei Chmutov
2008 arXiv   pre-print
We prove a relation between the signed Bollobas-Riordan polynomials of dual graphs.  ...  This relation unifies various recent results expressing the Jones polynomial of links as specializations of the Bollobas-Riordan polynomials.  ...  We express the Kauffman bracket (and hence the Jones polynomial) of L as a specialization of the Bollobás-Riordan polynomial of G s L .  ... 
arXiv:0711.3490v3 fatcat:hdhikjqsy5hupm7enyjflttf6i

Generalized duality for graphs on surfaces and the signed Bollobás–Riordan polynomial

Sergei Chmutov
2009 Journal of combinatorial theory. Series B (Print)  
We prove a relation between the signed Bollobás-Riordan polynomials of dual graphs.  ...  This relation unifies various recent results expressing the Jones polynomial of links as specializations of the Bollobás-Riordan polynomials.  ...  Chmutov for useful discussions and for showing me the signed version of the contraction-deletion property (Proposition 2.4) for the Bollobás-Riordan polynomial and to I. Moffatt and F.  ... 
doi:10.1016/j.jctb.2008.09.007 fatcat:bqc6ojir5fcb7aosabsgda2z4e

Expansions for the Bollobás-Riordan Polynomial of Separable Ribbon Graphs

Stephen Huggett, Iain Moffatt
2011 Annals of Combinatorics  
We give formulae for the Bollobas-Riordan polynomial of such a 2-decomposition, and derive the classical Brylawski formula for the Tutte polynomial of a tensor product as a (very) special case.  ...  We define 2-decompositions of ribbon graphs, which generalise 2-sums and tensor products of graphs.  ...  An expansion for the Bollobás-Riordan polynomial.  ... 
doi:10.1007/s00026-011-0116-3 fatcat:5jmloh74nfaw7lgyu53rruq27q

A recipe theorem for the topological Tutte polynomial of Bollobas and Riordan [article]

Joanna A. Ellis-Monaghan, Irasema Sarmiento
2009 arXiv   pre-print
We conclude by placing the results of Chumutov and Pak [The Kauffman bracket and the Bollobas-Riordan polynomial of ribbon graphs, Moscow Mathematical Journal 7(3) (2007) 409-418] for virtual links in  ...  Ann. 323, 81-96 (2002)], Bollobas and Riordan generalized the classical Tutte polynomial to graphs cellularly embedded in surfaces, i.e. ribbon graphs, thus encoding topological information not captured  ...  Acknowledgements We are grateful to Dan Archdeacon, Iain Moffat, and Lorenzo Traldi for a number of useful and interesting conversations.  ... 
arXiv:0903.2643v1 fatcat:4ni7elnnwnbbrnzkkj6tyu2hmu

Polynomial invariants of graphs on surfaces

Ross Askanazi, Sergei Chmutov, Charles Estill, Jonathan Michel, Patrick Stollenwerk
2013 Quantum Topology  
This will give an expression of the polynomial, defined by M.Las Vergnas in a combinatorial way using matroids as a specialization of the Krushkal polynomial, defined using the symplectic structure in  ...  For a graph embedded into a surface, we relate many combinatorial parameters of the cycle matroid of the graph and the bond matroid of the dual graph with the topological parameters of the embedding.  ...  and Bollobás-Riordan polynomials [Kru] .  ... 
doi:10.4171/qt/35 fatcat:hgao7lb2nrgwrgtjv7hvqxrei4

Distance Hereditary Graphs and the Interlace Polynomial [article]

Joanna A. Ellis-Monaghan, Irasema Sarmiento
2006 arXiv   pre-print
We also show a relation between the two-variable interlace polynomial and the topological Tutte polynomial of Bollobás and Riordan.  ...  These include relations between the interlace polynomial and the Tutte polynomial and the computational complexity of the vertex-nullity interlace polynomial.  ...  Acknowledgements: DH Graphs and the Interlace Polynomial. We would like to thank an anonymous referee for suggesting a number of productive areas of investigation, Dr.  ... 
arXiv:math/0604088v2 fatcat:frqcvonlafbfpf74kqy73k5nxq

Distance Hereditary Graphs and the Interlace Polynomial

JOANNA A. ELLIS-MONAGHAN, IRASEMA SARMIENTO
2007 Combinatorics, probability & computing  
We also show a relation between the two-variable interlace polynomial and the topological Tutte polynomial of Bollobás and Riordan in [BR01] .  ...  These include relations between the interlace polynomial and the Tutte polynomial and the computational complexity of the vertex-nullity interlace polynomial.  ...  Acknowledgements: DH Graphs and the Interlace Polynomial. We would like to thank an anonymous referee for suggesting a number of productive areas of investigation, Dr.  ... 
doi:10.1017/s0963548307008723 fatcat:zw5mr5ayqnhl5f7gzht7ttxkra

Topological graph polynomials and quantum field theory Part I: heat kernel theories

Thomas Krajewski, Vincent Rivasseau, Adrian Tanasă, Zhituo Wang
2010 Journal of Noncommutative Geometry  
versions of the Bollobás-Riordan polynomials.  ...  We show how the Symanzik polynomials of quantum field theory are particular multivariate versions of the Tutte polynomial, and how the new polynomials of noncommutative quantum field theory are special  ...  Ellis-Monaghan for introducing us to Bollobás-Riordan polynomials and Rȃzvan Gurȃu and Fabien Vignes-Tourneret for interesting discussions at an early stage of this work.  ... 
doi:10.4171/jncg/49 fatcat:l6hdj6sqard2fdtdcuhclxp2xu
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