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

James F. Geelen, A.M.H. Gerards, Geoff Whittle
2002 Journal of combinatorial theory. Series B (Print)  
We prove that a class of 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.  ... 
doi:10.1006/jctb.2001.2082 fatcat:owsrw6mjpngavhief37dum5mmy

Well-quasi-ordering in lattice path matroids [article]

Meenu Mariya Jose, Dillon Mayhew
2018 arXiv   pre-print
Though lattice path 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.  ... 
arXiv:1806.10260v1 fatcat:37vdwy3qqja6pdr4umhlk42gue

TOWARDS A MATROID-MINOR STRUCTURE THEORY [chapter]

Jim Geelen, Bert Gerards, Geoff Whittle
2007 Combinatorics, Complexity, and Chance  
For example, Robertson 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.  ... 
doi:10.1093/acprof:oso/9780198571278.003.0005 fatcat:rnrqxwsn6fdhdixknsck3bxqqy

Special issue in honor of Geoff Whittle

Dillon Mayhew, James Oxley, Charles Semple
2013 Advances in Applied Mathematics  
This collaboration has produced 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.  ... 
doi:10.1016/j.aam.2012.08.006 fatcat:54tqw7x7onf7jgqfl6f4vkopne

Branch-depth: Generalizing tree-depth of graphs [article]

Matt DeVos and O-joung Kwon and Sang-il Oum
2019 arXiv   pre-print
Furthermore we investigate various potential generalizations of tree-depth to 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.  ... 
arXiv:1903.11988v1 fatcat:gi4wqco2effijji7oqduxlh2qq

Rank-width and well-quasi-ordering of skew-symmetric or symmetric matrices

Sang-il Oum
2012 Linear Algebra and its Applications  
This generalizes three theorems on 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  ... 
doi:10.1016/j.laa.2011.09.027 fatcat:ocgmcqgwr5fofgqhliewxizody

Branch and Tree Decomposition Techniques for Discrete Optimization [chapter]

Illya V. Hicks, Arie M. C. A. Koster, Elif Kolotoğlu
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).  ... 
doi:10.1287/educ.1053.0017 fatcat:czyk6fbzdre53cp4zxfyckd5ga

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

Mamadou Moustapha Kanté
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''.  ... 
doi:10.1016/j.ejc.2012.03.034 fatcat:g3ivd3hytrcxjjwgsakwf3rzpq

Rank-width: Algorithmic and structural results

Sang-il Oum
2017 Discrete Applied Mathematics  
This survey aims to summarize known algorithmic 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.  ... 
doi:10.1016/j.dam.2016.08.006 fatcat:pvvs5yq5hvhxderfshfih3atti

Rank-Width and Well-Quasi-Ordering

Sang-il Oum
2008 SIAM Journal on Discrete Mathematics  
By extending their arguments, Geelen, Gerards, 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  ... 
doi:10.1137/050629616 fatcat:dfypq474zbatxoph7im6pfe3ty

Some Hard Problems on Matroid Spikes

Petr Hlineny
2007 Theory of Computing Systems  
Whittle, 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.  ... 
doi:10.1007/s00224-007-1307-5 fatcat:m2m4ooibbne2leibtycu5xd34y

Trees, grids, and MSO decidability: From graphs to matroids

Petr Hliněný, Detlef Seese
2006 Theoretical Computer Science  
Whittle, Branchwidth 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.  ... 
doi:10.1016/j.tcs.2005.10.006 fatcat:dtsh4w7hs5dbtkz6nyqjef3cp4

Graph minor theory

László Lovász
2005 Bulletin of the American Mathematical Society  
We describe the precise formulation of the main results 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.  ... 
doi:10.1090/s0273-0979-05-01088-8 fatcat:nwz23353z5bubdsyobrdls3ksu

On the Excluded Minors for Matroids of Branch-Width Three

Petr Hliněný
2002 Electronic Journal of Combinatorics  
We also construct numbers of such ternary 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  ... 
doi:10.37236/1648 fatcat:rzviyjy7cff2hnjnnzkutlkoti

Obstructions for bounded branch-depth in matroids [article]

J. Pascal Gollin, Kevin Hendrey, Dillon Mayhew, Sang-il Oum
2020 arXiv   pre-print
We prove that 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)]).  ... 
arXiv:2003.13975v1 fatcat:3ob47hcvtzaq5ccgg7opzfjgje
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