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Branch-Width and Well-Quasi-Ordering in Matroids and Graphs

2002
*
Journal of combinatorial theory. Series B (Print)
*

We prove that a class of

doi:10.1006/jctb.2001.2082
fatcat:owsrw6mjpngavhief37dum5mmy
*matroids*representable over a fixed finite field*and*with bounded*branch*-*width*is*well*-*quasi*-*ordered*under taking minors. ... With some extra work, the result implies Robertson*and*Seymour's result that*graphs*with bounded tree-*width*(or equivalently, bounded*branch*-*width*) are*well*-*quasi*-*ordered*under taking minors. ... For example, the relation ''A is isomorphic to a minor of B'' is a*quasi*-*order*on any set of*graphs*or*BRANCH*-*WIDTH**AND**WELL*-*QUASI*-*ORDERING**matroids*. ...##
###
Well-quasi-ordering in lattice path matroids
[article]

2018
*
arXiv
*
pre-print

Though lattice path

arXiv:1806.10260v1
fatcat:37vdwy3qqja6pdr4umhlk42gue
*matroids*are not*well*-*quasi*-*ordered*, we prove that lattice path*matroids*of bounded*branch*-*width*are*well*-*quasi*-*ordered*. ... Lattice path*matroids*form a subclass of transversal*matroids**and*were introduced by Bonin, de Mier*and*Noy. Transversal*matroids*are not*well*-*quasi*-*ordered*, even when the*branch*-*width*is restricted. ...*In*essence, a*graph*class or*matroid*class is*well*-*quasi*-*ordered*if*and*only if it does not have a bad sequence, as*graph**and**matroid*classes do not contain infinite decreasing sequences. ...##
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TOWARDS A MATROID-MINOR STRUCTURE THEORY
[chapter]

2007
*
Combinatorics, Complexity, and Chance
*

For example, Robertson

doi:10.1093/acprof:oso/9780198571278.003.0005
fatcat:rnrqxwsn6fdhdixknsck3bxqqy
*and*Seymour [21] prove that any class of*graphs*of bounded tree*width*is*well*-*quasi*-*ordered*so that G is*well*-*quasi*-*ordered*. ...*Branch**width*is equivalent to tree*width**in*that a class of*graphs*, or*matroids*, has bounded*branch**width*if*and*only if it has bounded tree*width*. ...##
###
Special issue in honor of Geoff Whittle

2013
*
Advances in Applied Mathematics
*

This collaboration has produced

doi:10.1016/j.aam.2012.08.006
fatcat:54tqw7x7onf7jgqfl6f4vkopne
*matroid*analogues of the*well*-*quasi*-*ordering*results from Robertson*and*Seymour's monumental*Graph*Minors Project. ... Tutte*and*Paul Seymour as the most significant ever done*in**matroid*theory.*In*this preface, we will introduce*matroids*after giving some biographical details of Geoff's life. ...*In*2009, Jim, Bert,*and*Geoff announced that they had proved that the class of binary*matroids*is*well*-*quasi*-*ordered*. ...##
###
Branch-depth: Generalizing tree-depth of graphs
[article]

2019
*
arXiv
*
pre-print

Furthermore we investigate various potential generalizations of tree-depth to

arXiv:1903.11988v1
fatcat:gi4wqco2effijji7oqduxlh2qq
*matroids**and*prove that*matroids*representable over a fixed finite field having no large circuits are*well*-*quasi*-*ordered*by ... For a*graph*G = (V,E)*and*a subset A of E we let λ_G (A) be the number of vertices incident with an edge*in*A*and*an edge*in*E ∖ A. ... circuits are*well*-*quasi*-*ordered*by the*matroid*restriction. ...##
###
Rank-width and well-quasi-ordering of skew-symmetric or symmetric matrices

2012
*
Linear Algebra and its Applications
*

This generalizes three theorems on

doi:10.1016/j.laa.2011.09.027
fatcat:ocgmcqgwr5fofgqhliewxizody
*well*-*quasi*-*ordering*of*graphs*or*matroids*admitting good tree-like decompositions; (1) Robertson*and*Seymour's theorem for*graphs*of bounded tree-*width*, (2) Geelen, Gerards ... ,*and*Whittle's theorem for*matroids*representable over a fixed finite field having bounded*branch*-*width*,*and*(3) Oum's theorem for*graphs*of bounded rank-*width*with respect to pivot-minors. ... [8] on wellquasi-*ordering*of F-representable*matroids*of bounded*branch*-*width*for a finite field F as*well*as the theorem by Robertson*and*Seymour [14] on*well*-*quasi*-*ordering*of*graphs*of bounded ...##
###
Branch and Tree Decomposition Techniques for Discrete Optimization
[chapter]

2005
*
Emerging Theory, Methods, and Applications
*

*in*monadic second

*order*logic. ... The algorithmic importance of

*branch*decompositions

*and*tree decompositions for solving NP-hard problems modelled on

*graphs*was first realized by computer scientists

*in*relation to formulating

*graph*problems ... the Netherlands Organization for Scientific Research (project Treewidth

*and*Combinatorial Optimization). ...

##
###
Well-quasi-ordering of matrices under Schur complement and applications to directed graphs

2012
*
European journal of combinatorics (Print)
*

*In*[Rank-

*Width*

*and*

*Well*-

*Quasi*-

*Ordering*of Skew-Symmetric or Symmetric Matrices, arXiv:1007.3807v1] Oum proved that, for a fixed finite field F, any infinite sequence M_1,M_2,... of (skew) symmetric matrices ... We generalise this result to σ-symmetric matrices introduced by Rao

*and*myself

*in*[The Rank-

*Width*of Edge-Coloured

*Graphs*, arXiv:0709.1433v4]. ... Courcelle

*and*the anonymous referee for their helpful comments. The author is supported by the DORSO project of ''Agence Nationale Pour la Recherche''. ...

##
###
Rank-width: Algorithmic and structural results

2017
*
Discrete Applied Mathematics
*

This survey aims to summarize known algorithmic

doi:10.1016/j.dam.2016.08.006
fatcat:pvvs5yq5hvhxderfshfih3atti
*and*structural results on rank-*width*of*graphs*. ... Rank-*width*is a*width*parameter of*graphs*describing whether it is possible to decompose a*graph*into a tree-like structure by 'simple' cuts. ... Kanté, Eun Jung Kim, O-joung Kwon,*and*Jisu Jeong for their helpful comments on an early draft of this paper. ...##
###
Rank-Width and Well-Quasi-Ordering

2008
*
SIAM Journal on Discrete Mathematics
*

By extending their arguments, Geelen, Gerards,

doi:10.1137/050629616
fatcat:dfypq474zbatxoph7im6pfe3ty
*and*Whittle (2002) proved that binary*matroids*of bounded*branch*-*width*are*well*-*quasi*-*ordered*by the*matroid*minor relation. ... We prove that*graphs*of bounded rank-*width*are*well*-*quasi*-*ordered*by the vertexminor relation;*in*other words, for every infinite sequence G 1 , G 2 , . . . of*graphs*of rank-*width*(or clique-*width*) at ... This work was performed while the author was at the Program*in*Applied*and*Computational Mathematics, Princeton University, Princeton, New Jersey*and*partially supported by the SRC program of Korea Science ...##
###
Some Hard Problems on Matroid Spikes

2007
*
Theory of Computing Systems
*

Whittle,

doi:10.1007/s00224-007-1307-5
fatcat:m2m4ooibbne2leibtycu5xd34y
*Branch*-*Width**and**Well*-*Quasi*-*Ordering**in**Matroids**and**Graphs*, J. Combin. Theory Ser. B 84 (2002), 270–290. 5. P. ... On the other hand, an interesting class of*matroids*of*branch*-*width*three, called spikes, is not*well*-*quasi*-*ordered*over any infinite field. ...##
###
Trees, grids, and MSO decidability: From graphs to matroids

2006
*
Theoretical Computer Science
*

Whittle, Branchwidth

doi:10.1016/j.tcs.2005.10.006
fatcat:dtsh4w7hs5dbtkz6nyqjef3cp4
*and**well*-*quasi*-*ordering**in**matroids**and**graphs*, J. Combin. Theory Ser. B 84 (2002) 270-290; J.F. Geelen, A.H.M. Gerards, N. Robertson, G.P. ... (*Matroids*can be viewed as a wide generalization of*graphs*,*and*they seem to capture some second*order*properties*in*a more suitable way than*graphs*themselves, cf. the recent development*in**matroid*structure ... Courcelle, the organizer of the workshop on "Logic*and**Graph*Transformations" at the ICGT 2004 conference, for inviting us to present our research results there. ...##
###
Graph minor theory

2005
*
Bulletin of the American Mathematical Society
*

We describe the precise formulation of the main results

doi:10.1090/s0273-0979-05-01088-8
fatcat:nwz23353z5bubdsyobrdls3ksu
*and*survey some of its applications to algorithmic*and*structural problems*in**graph*theory. ...*in*a tree-like fashion from*graphs*that can almost be embedded*in*a fixed surface. ... So Wagner's conjecture follows*in*this case if we show that for any positive integer k, the class of*graphs*with tree-*width*at most k is*well*-*quasi*-*ordered*by the minor relation. ...##
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On the Excluded Minors for Matroids of Branch-Width Three

2002
*
Electronic Journal of Combinatorics
*

We also construct numbers of such ternary

doi:10.37236/1648
fatcat:rzviyjy7cff2hnjnnzkutlkoti
*and*quaternary*matroids*$ M$,*and*provide a simple practical algorithm for finding a*width*-$3$*branch*-decomposition of a*matroid*. ... Whittle, On*Matroids*of*Branch*-*Width*Three, submitted 2001] that if $M$ is an excluded minor for*matroids*of*branch*-*width*$3$, then $ M$ has at most $14$ elements. ... Besides others, we want to mention the following recent works:*well*-*quasi*-*ordering*of*matroids*of bounded*branch*-*width*over finite fields [4] , sizebounds on the excluded minors for*matroids*of fixed ...##
###
Obstructions for bounded branch-depth in matroids
[article]

2020
*
arXiv
*
pre-print

We prove that

arXiv:2003.13975v1
fatcat:3ob47hcvtzaq5ccgg7opzfjgje
*matroids*with sufficiently large*branch*-depth either contain the cycle*matroid*of a large fan*graph*as a minor or have large*branch*-*width*. ... DeVos, Kwon,*and*Oum introduced the concept of*branch*-depth of*matroids*as a natural analogue of tree-depth of*graphs*. ... If G has*branch*-*width*at most w, then the*quasi*-graphic*matroid*M :" M pG, B, L, Fq has*branch*-*width*at most w`2. Theorem 5. 4 ( 4 Robertson*and*Seymour [18, (2.1)]*and*[19, (5.1)]). ...
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