DOCSLIB.ORG

Theory of Matroids and Applications

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Theory of Matroids and Applications THEORY OF MATROIDS AND APPLICATIONS JORGE RAMÍREZ ALFONSÍN Abstract. A matroid is a combinatorial structure that can be defined by keeping the main ‘set properties’ of the linear dependency in vector spaces. Matroids sat- isfy several equivalent axioms and have a fundamental notion of duality giving the right setting to study a number of classical problems. In this notes, we give a basic introduction of the theory of matroids as well as some applications. Contents 1. Introduction 1 2. Axioms 2 2.1. Independents 2 2.2. Circuits 2 2.3. Bases 3 2.4. Rank 4 2.5. Closure 4 2.6. Greedy Algorithm 5 3. Applications : combinatorial optimisation 5 3.1. Transversal matroid 5 3.2. Assigment Problem 5 4. Duality 6 4.1. Graphic matroids 7 4.2. Minors 7 5. Representable matroids 9 6. Regular matroids 10 7. Union of matroids 11 8. Tutte Polynomial 12 8.1. General properties 12 8.2. Chromatic Polynomial 12 8.3. Chromatic polynomial with defect 13 8.4. Flow Polynomial 13 8.5. Ehrhart polynomial 13 8.6. Acyclic and totally cyclic orientations 14 References 14 Partially supported by the ANR TEOMATRO grant ANR-10-BLAN 0207. 1 2 JORGE RAMÍREZ ALFONSÍN 1. Introduction The word of matroid appears for the first time in the fundamental paper due to Whit- ney in 1935 [16]. In this paper, the structure of a matroid was introduced as the set abstraction of the dependency relations between the column vectors of a matrix (explaining the suffix ‘oid’ indicating that is a structure coming from a matrix). The applications of matroids have their origins in the area of combinatorics (graphs, dis- crete optimisation, polytopes, etc.). For the last thirty years, many other applications of matroid have appeared in many other differents areas as algebraic geometry, stratifi- cation of grassmanniennes, discrete geometry, etc. We recommend the reader the book [9] for further details on matroid theory. 2. Axioms Matroids can be characterized by various different equivalent axioms. 2.1. Independents. A matroid M is an ordered couple (E; I) where E is a finite set (E = f1; : : : ; ng) and I is a collection of subsets of E verifying the following (I1) ; 2 I, (I2) If I1 2 I and I2 ⊂ I1 then I2 2 I, (I3) (augmentation property) If I1;I2 2 I and jI1j < jI2j then there exist e 2 I2nI1 such that I1 [ e 2 I. The members of I are called the independents of M. A subset of E that is not in I is called dependent. Example 2.1. Let fe1; : : : ; eng be the set of column vectors of a matrix with coefficients in a field F. Let I be the collection of all the subsets of fi1; : : : ; img ⊆ f1; : : : ; ng = E such that the set of columns fei1 ; : : : ; eim g are linearly independente on F. Then, (E; I) is a matroid. A matroid that is isomorpic to a matroid obtained from a matrix over a field F is called representable ou linear over F. Example 2.2. Let A be the following matrix with coefficients in R 1 2 3 4 5 A = 1 0 0 1 1 0 1 0 0 1 We have that I(M) = f;; f1g; f2g; f3g; f4g; f1; 2g; f1; 3g; f2; 4g; f3; 4g; f2; 3gg. 2.2. Circuits. A subset X ⊆ E is said to be minimal dependent if any subset of X, different from X, is independent. A minimal dependent subset of a matroid M is called a circuit of M. We denote by C the set of circuits of a matroid. We observe that I can be determined by C (the membres of I are all the subsets of E not containing a member of C). A circuit of cardinality one is called loop. We have that C is a set of circuits of a matroid on E if and only if C verifies the following properties : (C1) ; 62 C, (C2) C1;C2 2 C and C1 ⊆ C2 then C1 = C2, (C3) (elimination property) If C1;C2 2 C;C1 6= C2 and e 2 C1 \ C2 then there exists C3 2 C such that C3 ⊆ fC1 [ C2gnfeg. THEORY OF MATROIDS AND APPLICATIONS 3 Example 2.3. In the graph G = (V; E), we call cycle the set of edges in a cycle of G, and simple cycle if such a set is minimal by inclusion. Here, we consider only simple cycles. Let G = (V; E) be a graph and let C be the set of cycles of G. Then, C is the set of circuits of a matroid on E, denoted by M(G). A matroid obtained on this way is called graphic matroid. e5 e e e1 4 3 e2 Figure 1. Graph with 3 verticess Remark 2.4. There is not the notion of vertex in a matroid. The matroid associated to a graph do not determine, in general, the graph. For instance, any connected graph without cycle has the same matroid. The associated matroid determine the graph if and only if the graph is 3-connected. Example 2.5. Let M(G) be the graphic matroid obtained from graph G in Figure 1, and let A be the matrix over R given in Exemple 2.2. We can verify that M(G) is isomorphic to M(A) (under the bijection ei ! i). Remark 2.6. It turns out that a graphic matroid is always linear. Let G = (V; E) be a graph and fxi; i 2 V g the canonique bases of a vector space over an arbitrary field. We can verify that, as in Example 2.5, the graph G = (V; E) is associated to the same matroid as the set of vectors xi − xj pour (i; j) 2 E. Example 2.7. Let M(G0) be the graphic matroid obtained from graph G0 in Figure 2. Let A0 be the matrix obtained by following the construction of Remark 2.6. x1 x2 x3 x4 0 1 1 0 0 1 A0 = B −1 0 1 0 C B C @ 0 −1 −1 1 A 0 0 0 −1 0 0 We can check that M(G ) is isomorphic to M(A ) (under the bijection xi ! i). Notice that the cycle formed by the edges a = f1; 2g; b = f1; 3g et c = f2; 3g in the graph G0 correspond to the linear dependency x2 − x1 = x3. 2.3. Bases. A base of a matroid is a independ set maximal by inclusion. All the bases of a matroid have the same cardinality (the same number of elements). Indeed, otherwise, we would have two bases B1;B2 with jB1j < jB2j and so, by (I3) there exist x 2 B2nB1 such that B1 [ x 2 I which is a contradiction since B1 is a maximal independent. The collection B of bases verify the following conditions (B1) B 6= ;. 4 JORGE RAMÍREZ ALFONSÍN 1 a b c d 2 3 4 Figure 2. Graph G0 (B2) (exchange property) B1;B2 2 B and x 2 B1nB2 then there exist y 2 B2nB1 such that (B1nx) [ y 2 B. If I is the collection of subsets contained in one of the members of B then (E; I) is a matroid. Remark 2.8. If G is a connected graph then the bases of M(G) correspond to the set of all spanning trees of G. 2.4. Rank. Let M = (E; I) and X ⊆ E. The rank of X, denoted by rM (X), is the cardinality of the largest independent contained in X, that is, rM (X) = maxfjY j : Y ⊆ X; Y 2 Ig: Equivalently, we define the set InX = fI ⊆ XjI 2 Ig. Then, (X; InX) is a matroid, denoted by MjX and called restriction of M to X. The rank rM (X) of X is the cardinality of a base in MjX . It can be proved that r = rM is the rank function of a matroid (E; I) where I = fI ⊆ E : r(I) = jIjg; if and only if r verify the following conditions (R1)0 ≤ r(X) ≤ jXj; for all X ⊆ E, (R2) r(X) ≤ r(Y ), for all X ⊆ Y , (R3) (sub-modularity) r(X [ Y ) + r(X \ Y ) ≤ r(X) + r(Y ) for all X; Y ⊂ E. 2.5. Closure. The closure of a set X in M is defined by cl(X) = fx 2 Ejr(X [ x) = r(X)g: It can be proved that the function cl : P(E) !P(E) is the closure function of a matroid if and only if cl verify the following conditions (CL1) (extensivity) If X ⊆ E then X ⊆ cl(X). (CL2) (increasing) If X ⊆ Y ⊆ E then cl(X) ⊆ cl(Y ). (CL3) (indempotent) If X ⊆ E then cl(cl(X)) = cl(X). (CL4) (exchange property) If X ⊆ E; x 2 E and y 2 cl(X [ x) − cl(X) then x 2 cl(X [ y). Let X ⊂ E, cl(X) is also called flat of X. X is said to be closed if X = cl(X). The set E is a closed set of rank rM . The rank 0 closed sets are formed by the loops of M.A closed set of rank 1 or point is the class of parallel elements. A matroid such that ; is a closed set and all its points contain only one element, is called simple.A hyperplan is a closed set of rank rM − 1. THEORY OF MATROIDS AND APPLICATIONS 5 Example 2.9. Let M(G0) be a graphic matroid obtained from graph G0 in Figure 2. It can be verified that : r(fa; b; cg) = r(fc; dg) = r(fa; dg) = 2 and clfa; bg = fa; b; cg.
Load more

Recommended publications
  • Duality Respecting Representations and Compatible Complexity
    Duality Respecting Representations and Compatible Complexity Measures for Gammoids Immanuel Albrecht FernUniversit¨at in Hagen, Department of Mathematics and Computer Science, Discrete Mathematics and Optimization D-58084 Hagen, Germany Abstract We show that every gammoid has special digraph representations, such that a representation of the dual of the gammoid may be easily obtained by reversing all arcs. In an informal sense, the duality notion of a poset applied to the digraph of a special representation of a gammoid commutes with the operation of forming the dual of that gammoid. We use these special representations in order to define a complexity measure for gammoids, such that the classes of gammoids with bounded complexity are closed under duality, minors, and direct sums. Keywords: gammoids, digraphs, duality, complexity measure 2010 MSC: 05B35, 05C20, 06D50 A well-known result due to J.H. Mason is that the class of gammoids is closed under duality, minors, and direct sums [1]. Furthermore, it has been shown by D. Mayhew that every gammoid is also a minor of an excluded minor for the class of gammoids [2], which indicates that handling the class of all gammoids may get very involved. In this work, we introduce a notion of complexity for gammoids which may be used to define subclasses of gammoids with bounded complexity, that still have the desirable property of being closed under duality, minors, and direct sums; yet their representations have a more limited number arXiv:1807.00588v2 [math.CO] 27 Jun 2020 of arcs than the general class of gammoids. 1. Preliminaries In this work, we consider matroids to be pairs M = (E, I) where E is a finite set and I is a system of independent subsets of E subject to the usual axioms ([3], Sec.
    [Show full text]
  • The Minimum Cost Query Problem on Matroids with Uncertainty Areas Arturo I
    The minimum cost query problem on matroids with uncertainty areas Arturo I. Merino Department of Mathematical Engineering and CMM, Universidad de Chile & UMI-CNRS 2807, Santiago, Chile [email protected] José A. Soto Department of Mathematical Engineering and CMM, Universidad de Chile & UMI-CNRS 2807, Santiago, Chile [email protected] Abstract We study the minimum weight basis problem on matroid when elements’ weights are uncertain. For each element we only know a set of possible values (an uncertainty area) that contains its real weight. In some cases there exist bases that are uniformly optimal, that is, they are minimum weight bases for every possible weight function obeying the uncertainty areas. In other cases, computing such a basis is not possible unless we perform some queries for the exact value of some elements. Our main result is a polynomial time algorithm for the following problem. Given a matroid with uncertainty areas and a query cost function on its elements, find the set of elements of minimum total cost that we need to simultaneously query such that, no matter their revelation, the resulting instance admits a uniformly optimal base. We also provide combinatorial characterizations of all uniformly optimal bases, when one exists; and of all sets of queries that can be performed so that after revealing the corresponding weights the resulting instance admits a uniformly optimal base. 2012 ACM Subject Classification Mathematics of computing → Matroids and greedoids Keywords and phrases Minimum spanning tree, matroids, uncertainty, queries Funding Work supported by Conicyt via Fondecyt 1181180 and PIA AFB-170001. 1 Introduction We study fundamental combinatorial optimization problems on weighted structures where the numerical data is uncertain but it can be queried at a cost.
    [Show full text]
  • The Tutte Polynomial of Some Matroids Arxiv:1203.0090V1 [Math
    The Tutte polynomial of some matroids Criel Merino∗, Marcelino Ram´ırez-Iba~nezy Guadalupe Rodr´ıguez-S´anchezz March 2, 2012 Abstract The Tutte polynomial of a graph or a matroid, named after W. T. Tutte, has the important universal property that essentially any mul- tiplicative graph or network invariant with a deletion and contraction reduction must be an evaluation of it. The deletion and contraction operations are natural reductions for many network models arising from a wide range of problems at the heart of computer science, engi- neering, optimization, physics, and biology. Even though the invariant is #P-hard to compute in general, there are many occasions when we face the task of computing the Tutte polynomial for some families of graphs or matroids. In this work we compile known formulas for the Tutte polynomial of some families of graphs and matroids. Also, we give brief explanations of the techniques that were use to find the for- mulas. Hopefully, this will be useful for researchers in Combinatorics arXiv:1203.0090v1 [math.CO] 1 Mar 2012 and elsewhere. ∗Instituto de Matem´aticas,Universidad Nacional Aut´onomade M´exico,Area de la Investigaci´onCient´ıfica, Circuito Exterior, C.U. Coyoac´an04510, M´exico,D.F.M´exico. e-mail:[email protected]. Supported by Conacyt of M´exicoProyect 83977 yEscuela de Ciencias, Universidad Aut´onoma Benito Ju´arez de Oaxaca, Oaxaca, M´exico.e-mail:[email protected] zDepartamento de Ciencias B´asicas Universidad Aut´onoma Metropolitana, Az- capozalco, Av. San Pablo No. 180, Col. Reynosa Tamaulipas, Azcapozalco 02200, M´exico D.F.
    [Show full text]
  • Lecture 0: Matroid Basics
    Parameterized Algorithms using Matroids Lecture 0: Matroid Basics Saket Saurabh The Institute of Mathematical Sciences, India and University of Bergen, Norway. ADFOCS 2013, MPI, August 04-09, 2013 Kruskal's Greedy Algorithm for MWST Let G = (V; E) be a connected undirected graph and let ≥0 w : E ! R be a weight function on the edges. Kruskal's so-called greedy algorithm is as follows. The algorithm consists of selecting successively edges e1; e2; : : : ; er. If edges e1; e2; : : : ; ek has been selected, then an edge e 2 E is selected so that: 1 e=2f e1; : : : ; ekg and fe; e1; : : : ; ekg is a forest. 2 w(e) is as small as possible among all edges e satisfying (1). We take ek+1 := e. If no e satisfying (1) exists then fe1; : : : ; ekg is a spanning tree. Kruskal's Greedy Algorithm for MWST Let G = (V; E) be a connected undirected graph and let ≥0 w : E ! R be a weight function on the edges. Kruskal's so-called greedy algorithm is as follows. The algorithm consists of selecting successively edges e1; e2; : : : ; er. If edges e1; e2; : : : ; ek has been selected, then an edge e 2 E is selected so that: 1 e=2f e1; : : : ; ekg and fe; e1; : : : ; ekg is a forest. 2 w(e) is as small as possible among all edges e satisfying (1). We take ek+1 := e. If no e satisfying (1) exists then fe1; : : : ; ekg is a spanning tree. It is obviously not true that such a greedy approach would lead to an optimal solution for any combinatorial optimization problem.
    [Show full text]
  • Branch-Depth: Generalizing Tree-Depth of Graphs
    Branch-depth: Generalizing tree-depth of graphs ∗1 †‡23 34 Matt DeVos , O-joung Kwon , and Sang-il Oum† 1Department of Mathematics, Simon Fraser University, Burnaby, Canada 2Department of Mathematics, Incheon National University, Incheon, Korea 3Discrete Mathematics Group, Institute for Basic Science (IBS), Daejeon, Korea 4Department of Mathematical Sciences, KAIST, Daejeon, Korea [email protected], [email protected], [email protected] November 5, 2020 Abstract We present a concept called the branch-depth of a connectivity function, that generalizes the tree-depth of graphs. Then we prove two theorems showing that this concept aligns closely with the no- tions of tree-depth and shrub-depth of graphs as follows. For a graph G = (V, E) and a subset A of E we let λG(A) be the number of vertices incident with an edge in A and an edge in E A. For a subset X of V , \ let ρG(X) be the rank of the adjacency matrix between X and V X over the binary field. We prove that a class of graphs has bounded\ tree-depth if and only if the corresponding class of functions λG has arXiv:1903.11988v2 [math.CO] 4 Nov 2020 bounded branch-depth and similarly a class of graphs has bounded shrub-depth if and only if the corresponding class of functions ρG has bounded branch-depth, which we call the rank-depth of graphs. 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 restriction.
    [Show full text]
  • A Dual Fano, and Dual Non-Fano Matroidal Network Stephen Lee Johnson California State University - San Bernardino, [email protected]
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by CSUSB ScholarWorks California State University, San Bernardino CSUSB ScholarWorks Electronic Theses, Projects, and Dissertations Office of Graduate Studies 6-2016 A Dual Fano, and Dual Non-Fano Matroidal Network Stephen Lee Johnson California State University - San Bernardino, [email protected] Follow this and additional works at: http://scholarworks.lib.csusb.edu/etd Part of the Other Mathematics Commons Recommended Citation Johnson, Stephen Lee, "A Dual Fano, and Dual Non-Fano Matroidal Network" (2016). Electronic Theses, Projects, and Dissertations. Paper 340. This Thesis is brought to you for free and open access by the Office of Graduate Studies at CSUSB ScholarWorks. It has been accepted for inclusion in Electronic Theses, Projects, and Dissertations by an authorized administrator of CSUSB ScholarWorks. For more information, please contact [email protected]. A Dual Fano, and Dual Non-Fano Matroidal Network A Thesis Presented to the Faculty of California State University, San Bernardino In Partial Fulfillment of the Requirements for the Degree Master of Arts in Mathematics by Stephen Lee Johnson June 2016 A Dual Fano, and Dual Non-Fano Matroidal Network A Thesis Presented to the Faculty of California State University, San Bernardino by Stephen Lee Johnson June 2016 Approved by: Dr. Chris Freiling, Committee Chair Date Dr. Wenxiang Wang, Committee Member Dr. Jeremy Aikin, Committee Member Dr. Charles Stanton, Chair, Dr. Corey Dunn Department of Mathematics Graduate Coordinator, Department of Mathematics iii Abstract Matroidal networks are useful tools in furthering research in network coding. They have been used to show the limitations of linear coding solutions.
    [Show full text]
  • Arxiv:2008.03027V2 [Math.CO] 19 Nov 2020 an Optimal Way
    A Whitney type theorem for surfaces: characterising graphs with locally planar embeddings Johannes Carmesin University of Birmingham November 20, 2020 Abstract We prove that for any parameter r an r-locally 2-connected graph G embeds r-locally planarly in a surface if and only if a certain ma- troid associated to the graph G is co-graphic. This extends Whitney's abstract planar duality theorem from 1932. 1 Introduction A fundamental question in Structural Graph Theory is how to embed graphs in surfaces. There are two main lines of research. Firstly, in the specific embedding problem we have a fixed surface and are interested in embedding a given graph in that surface. Mohar proved the existence of a linear time algorithm for this problem, solving the al- gorithmic aspect of this problem [16]. This proof was later simplified by Kawarabayashi, Mohar and Reed [13]. Secondly, in the general embedding problem, we are interested in finding for a given graph a surface so that the graph embeds in that surface in arXiv:2008.03027v2 [math.CO] 19 Nov 2020 an optimal way. Usually people used minimum genus as the optimality criterion [13, 15, 17]. However, in 1989 Thomassen showed that with this interpretation, the problem would be NP-hard [24]. Instead of minimum genus, here we use local planarity as our optimality criterion { and provide a polynomial algorithm for the general embedding problem. While this discrepancy in the algorithmic complexity implies that maxi- mally locally planar embeddings cannot always be of minimum genus, Thomassen 1 showed that they are of minimum genus if all face boundaries are shorter than non-contractible cycles of the embedding [25].
    [Show full text]
  • A Dual Fano, and Dual Non-Fano Matroidal Network
    California State University, San Bernardino CSUSB ScholarWorks Electronic Theses, Projects, and Dissertations Office of aduateGr Studies 6-2016 A Dual Fano, and Dual Non-Fano Matroidal Network Stephen Lee Johnson California State University - San Bernardino Follow this and additional works at: https://scholarworks.lib.csusb.edu/etd Part of the Other Mathematics Commons Recommended Citation Johnson, Stephen Lee, "A Dual Fano, and Dual Non-Fano Matroidal Network" (2016). Electronic Theses, Projects, and Dissertations. 340. https://scholarworks.lib.csusb.edu/etd/340 This Thesis is brought to you for free and open access by the Office of aduateGr Studies at CSUSB ScholarWorks. It has been accepted for inclusion in Electronic Theses, Projects, and Dissertations by an authorized administrator of CSUSB ScholarWorks. For more information, please contact [email protected]. A Dual Fano, and Dual Non-Fano Matroidal Network A Thesis Presented to the Faculty of California State University, San Bernardino In Partial Fulfillment of the Requirements for the Degree Master of Arts in Mathematics by Stephen Lee Johnson June 2016 A Dual Fano, and Dual Non-Fano Matroidal Network A Thesis Presented to the Faculty of California State University, San Bernardino by Stephen Lee Johnson June 2016 Approved by: Dr. Chris Freiling, Committee Chair Date Dr. Wenxiang Wang, Committee Member Dr. Jeremy Aikin, Committee Member Dr. Charles Stanton, Chair, Dr. Corey Dunn Department of Mathematics Graduate Coordinator, Department of Mathematics iii Abstract Matroidal networks are useful tools in furthering research in network coding. They have been used to show the limitations of linear coding solutions. In this paper we examine the basic information on network coding and matroid theory.
    [Show full text]
  • The Circuit and Cocircuit Lattices of a Regular Matroid
    University of Vermont ScholarWorks @ UVM Graduate College Dissertations and Theses Dissertations and Theses 2020 The circuit and cocircuit lattices of a regular matroid Patrick Mullins University of Vermont Follow this and additional works at: https://scholarworks.uvm.edu/graddis Part of the Mathematics Commons Recommended Citation Mullins, Patrick, "The circuit and cocircuit lattices of a regular matroid" (2020). Graduate College Dissertations and Theses. 1234. https://scholarworks.uvm.edu/graddis/1234 This Thesis is brought to you for free and open access by the Dissertations and Theses at ScholarWorks @ UVM. It has been accepted for inclusion in Graduate College Dissertations and Theses by an authorized administrator of ScholarWorks @ UVM. For more information, please contact [email protected]. The Circuit and Cocircuit Lattices of a Regular Matroid A Thesis Presented by Patrick Mullins to The Faculty of the Graduate College of The University of Vermont In Partial Fulfillment of the Requirements for the Degree of Master of Science Specializing in Mathematical Sciences May, 2020 Defense Date: March 25th, 2020 Dissertation Examination Committee: Spencer Backman, Ph.D., Advisor Davis Darais, Ph.D., Chairperson Jonathan Sands, Ph.D. Cynthia J. Forehand, Ph.D., Dean of Graduate College Abstract A matroid abstracts the notions of dependence common to linear algebra, graph theory, and geometry. We show the equivalence of some of the various axiom systems which define a matroid and examine the concepts of matroid minors and duality before moving on to those matroids which can be represented by a matrix over any field, known as regular matroids. Placing an orientation on a regular matroid M allows us to define certain lattices (discrete groups) associated to M.
    [Show full text]
  • On the Interplay Between Graphs and Matroids
    On the interplay between graphs and matroids James Oxley Abstract “If a theorem about graphs can be expressed in terms of edges and circuits only it probably exemplifies a more general theorem about ma- troids.” This assertion, made by Tutte more than twenty years ago, will be the theme of this paper. In particular, a number of examples will be given of the two-way interaction between graph theory and matroid theory that enriches both subjects. 1 Introduction This paper aims to be accessible to those with no previous experience of matroids; only some basic familiarity with graph theory and linear algebra will be assumed. In particular, the next section introduces matroids by showing how such objects arise from graphs. It then presents a minimal amount of the- ory to make the rest of the paper comprehensible. Throughout, the emphasis is on the links between graphs and matroids. Section 3 begins by showing how 2-connectedness for graphs extends natu- rally to matroids. It then indicates how the number of edges in a 2-connected loopless graph can be bounded in terms of the circumference and the size of a largest bond. The main result of the section extends this graph result to ma- troids. The results in this section provide an excellent example of the two-way interaction between graph theory and matroid theory. In order to increase the accessibility of this paper, the matroid technicalities have been kept to a minimum. Most of those that do arise have been separated from the rest of the paper and appear in two separate sections, 4 and 10, which deal primarily with proofs.
    [Show full text]
  • Branchwidth of Graphic Matroids. Frédéric Mazoit, Stéphan Thomassé
    Branchwidth of graphic matroids. Frédéric Mazoit, Stéphan Thomassé To cite this version: Frédéric Mazoit, Stéphan Thomassé. Branchwidth of graphic matroids.. 2005. hal-00012312 HAL Id: hal-00012312 https://hal.archives-ouvertes.fr/hal-00012312 Preprint submitted on 19 Oct 2005 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Branchwidth of graphic matroids. Fr´ed´eric Mazoit∗ and St´ephan Thomass´e† Abstract Answering a question of Geelen, Gerards, Robertson and Whittle [2], we prove that the branchwidth of a bridgeless graph is equal to the branch- width of its cycle matroid. Our proof is based on branch-decompositions of hypergraphs. 1 Introduction. Let H = (V, E) be a hypergraph and (E1,E2) be a partition of E. The border of (E1,E2) is the set of vertices δ(E1,E2) which belong to both an edge of E1 and an edge of E2. We often write it δ(E1,E2), or often simply δ(E1). A component of E is a minimum nonempty subset C ⊆ E such that δ(C) = ∅. Let F be a subset of E. We denote by c(F ) the number of components of the subhypergraph of H spanned by F , i.e.
    [Show full text]
  • Matroids Lecturer: Michel X. Goemans Scribe: Bridget Eileen Tenner
    18.997 Topics in Combinatorial Optimization 9 March 2004 Matroids Lecturer: Michel X. Goemans Scribe: Bridget Eileen Tenner For reference, see Chapter 39 of Schrijver’s book. Definition 1Amatroid M =(S, I) is a finite groundS together set with a collection of sets I⊆2S,knownastheindependent sets, satisfying the following axioms: 1.IfI ∈I andJ ⊆ I thenJ ∈I. 2.IfI,J ∈I and|J| > |I|, then there exists anz element ∈ J \ I such thatI + z ∈I,where+ indicates union. If I ⊆ S is not an element of I, I is called a dependent set.Anelement I ∈I is a basisif it is a maximal (inclusion-wise) independent set. A circuitC of M is a minimal dependent set. A set I ∈I is a spanning setif I contains a basis. Example 1Linear matroids. LetA be anm-by-n matrix. LetS = {1,...,n} be the index set of the columnsA.I ⊆ S of is independent if the columns indexedI are by linearly independent. It is straightforward to check that the axioms for a matroid hold in this example. Observation 1All bases of a matroid have the same cardinality by the second axiom. For T ⊆ S,let the rankof T be rM (T )= r(T )=max{|I| : I ⊆ T and I ∈I}. Note that this is a generalization of the linear algebra definition of rank. Letting the set T equal S, we find that rM (S)= |B|, the size of a basis of the matroid. If F is a field, a matroid is said to be representableover F if the matrix A and linear independence are taken over F .
    [Show full text]