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

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. ...

##
###
Complexity of the Bollobás-Riordan Polynomial
[chapter]

*
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. ...

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

2011
*
Bulletin of the London Mathematical Society
*

We generalize

doi:10.1112/blms/bdr034
fatcat:7acylrioonbanh5yqrmocwapuy
*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*. ...##
###
Generalization of the Bollobás-Riordan polynomial for tensor graphs

2011
*
Journal of Mathematical Physics
*

This

doi:10.1063/1.3605312
fatcat:koqwkc5j6fap3cnyvdxo73yhfq
*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. ...##
###
Topological Tutte Polynomial
[article]

2017
*
arXiv
*
pre-print

This is a survey recent works on topological extensions

arXiv:1708.08132v1
fatcat:v22y3mf42nfpjfus3k7sgtxdr4
*of**the*Tutte*polynomial*. ...*The*arrow version*of**the**Bollobás*-*Riordan**polynomial*. ...*The*dichromatic version*of**the**Bollobás*-*Riordan**polynomial*. ...##
###
The Jones polynomial and graphs on surfaces

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*. ...

##
###
The chromatic polynomial of fatgraphs and its categorification

2008
*
Advances in Mathematics
*

We extend our construction and categorify

doi:10.1016/j.aim.2007.11.016
fatcat:dkwh42hzdrh2zhrxl4reficc7i
*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. ...##
###
Generalized duality for graphs on surfaces and the signed Bollobas-Riordan polynomial
[article]

2008
*
arXiv
*
pre-print

We prove a relation between

arXiv:0711.3490v3
fatcat:hdhikjqsy5hupm7enyjflttf6i
*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 . ...##
###
Generalized duality for graphs on surfaces and the signed Bollobás–Riordan polynomial

2009
*
Journal of combinatorial theory. Series B (Print)
*

We prove a relation between

doi:10.1016/j.jctb.2008.09.007
fatcat:bqc6ojir5fcb7aosabsgda2z4e
*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. ...##
###
Expansions for the Bollobás-Riordan Polynomial of Separable Ribbon Graphs

2011
*
Annals of Combinatorics
*

We give formulae for

doi:10.1007/s00026-011-0116-3
fatcat:5jmloh74nfaw7lgyu53rruq27q
*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*. ...##
###
A recipe theorem for the topological Tutte polynomial of Bollobas and Riordan
[article]

2009
*
arXiv
*
pre-print

We conclude by placing

arXiv:0903.2643v1
fatcat:4ni7elnnwnbbrnzkkj6tyu2hmu
*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. ...##
###
Polynomial invariants of graphs on surfaces

2013
*
Quantum Topology
*

This will give an expression

doi:10.4171/qt/35
fatcat:hgao7lb2nrgwrgtjv7hvqxrei4
*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] . ...##
###
Distance Hereditary Graphs and the Interlace Polynomial
[article]

2006
*
arXiv
*
pre-print

We also show a relation between

arXiv:math/0604088v2
fatcat:frqcvonlafbfpf74kqy73k5nxq
*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. ...##
###
Distance Hereditary Graphs and the Interlace Polynomial

2007
*
Combinatorics, probability & computing
*

We also show a relation between

doi:10.1017/s0963548307008723
fatcat:zw5mr5ayqnhl5f7gzht7ttxkra
*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. ...##
###
Topological graph polynomials and quantum field theory Part I: heat kernel theories

2010
*
Journal of Noncommutative Geometry
*

versions

doi:10.4171/jncg/49
fatcat:l6hdj6sqard2fdtdcuhclxp2xu
*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. ...
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