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Some Topics Concerning Graphs, Signed Graphs and Matroids
SOME TOPICS CONCERNING GRAPHS, SIGNED GRAPHS AND MATROIDS DISSERTATION Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of the Ohio State University By Vaidyanathan Sivaraman, M.S. Graduate Program in Mathematics The Ohio State University 2012 Dissertation Committee: Prof. Neil Robertson, Advisor Prof. Akos´ Seress Prof. Matthew Kahle ABSTRACT We discuss well-quasi-ordering in graphs and signed graphs, giving two short proofs of the bounded case of S. B. Rao's conjecture. We give a characterization of graphs whose bicircular matroids are signed-graphic, thus generalizing a theorem of Matthews from the 1970s. We prove a recent conjecture of Zaslavsky on the equality of frus- tration number and frustration index in a certain class of signed graphs. We prove that there are exactly seven signed Heawood graphs, up to switching isomorphism. We present a computational approach to an interesting conjecture of D. J. A. Welsh on the number of bases of matroids. We then move on to study the frame matroids of signed graphs, giving explicit signed-graphic representations of certain families of matroids. We also discuss the cycle, bicircular and even-cycle matroid of a graph and characterize matroids arising as two different such structures. We study graphs in which any two vertices have the same number of common neighbors, giving a quick proof of Shrikhande's theorem. We provide a solution to a problem of E. W. Dijkstra. Also, we discuss the flexibility of graphs on the projective plane. We conclude by men- tioning partial progress towards characterizing signed graphs whose frame matroids are transversal, and some miscellaneous results. -
Quasipolynomial Representation of Transversal Matroids with Applications in Parameterized Complexity
Quasipolynomial Representation of Transversal Matroids with Applications in Parameterized Complexity Daniel Lokshtanov1, Pranabendu Misra2, Fahad Panolan1, Saket Saurabh1,2, and Meirav Zehavi3 1 University of Bergen, Bergen, Norway. {daniello,pranabendu.misra,fahad.panolan}@ii.uib.no 2 The Institute of Mathematical Sciences, HBNI, Chennai, India. [email protected] 3 Ben-Gurion University, Beersheba, Israel. [email protected] Abstract Deterministic polynomial-time computation of a representation of a transversal matroid is a longstanding open problem. We present a deterministic computation of a so-called union rep- resentation of a transversal matroid in time quasipolynomial in the rank of the matroid. More precisely, we output a collection of linear matroids such that a set is independent in the trans- versal matroid if and only if it is independent in at least one of them. Our proof directly implies that if one is interested in preserving independent sets of size at most r, for a given r ∈ N, but does not care whether larger independent sets are preserved, then a union representation can be computed deterministically in time quasipolynomial in r. This consequence is of independent interest, and sheds light on the power of union representation. Our main result also has applications in Parameterized Complexity. First, it yields a fast computation of representative sets, and due to our relaxation in the context of r, this computation also extends to (standard) truncations. In turn, this computation enables to efficiently solve various problems, such as subcases of subgraph isomorphism, motif search and packing problems, in the presence of color lists. Such problems have been studied to model scenarios where pairs of elements to be matched may not be identical but only similar, and color lists aim to describe the set of compatible elements associated with each element. -
Partial Fields and Matroid Representation
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by UC Research Repository Partial Fields and Matroid Representation Charles Semple and Geoff Whittle Department of Mathematics Victoria University PO Box 600 Wellington New Zealand April 10, 1995 Abstract A partial field P is an algebraic structure that behaves very much like a field except that addition is a partial binary operation, that is, for some a; b ∈ P, a + b may not be defined. We develop a theory of matroid representation over partial fields. It is shown that many im- portant classes of matroids arise as the class of matroids representable over a partial field. The matroids representable over a partial field are closed under standard matroid operations such as the taking of mi- nors, duals, direct sums and 2{sums. Homomorphisms of partial fields are defined. It is shown that if ' : P1 → P2 is a non-trivial partial field homomorphism, then every matroid representable over P1 is rep- resentable over P2. The connection with Dowling group geometries is examined. It is shown that if G is a finite abelian group, and r>2, then there exists a partial field over which the rank{r Dowling group geometry is representable if and only if G has at most one element of order 2, that is, if G is a group in which the identity has at most two square roots. 1 1 Introduction It follows from a classical (1958) result of Tutte [19] that a matroid is rep- resentable over GF (2) and some field of characteristic other than 2 if and only if it can be represented over the rationals by the columns of a totally unimodular matrix, that is, by a matrix over the rationals all of whose non- zero subdeterminants are in {1; −1}. -
Parameterized Algorithms Using Matroids Lecture I: Matroid Basics and Its Use As Data Structure
Parameterized Algorithms using Matroids Lecture I: Matroid Basics and its use as data structure Saket Saurabh The Institute of Mathematical Sciences, India and University of Bergen, Norway, ADFOCS 2013, MPI, August 5{9, 2013 1 Introduction and Kernelization 2 Fixed Parameter Tractable (FPT) Algorithms For decision problems with input size n, and a parameter k, (which typically is the solution size), the goal here is to design an algorithm with (1) running time f (k) nO , where f is a function of k alone. · Problems that have such an algorithm are said to be fixed parameter tractable (FPT). 3 A Few Examples Vertex Cover Input: A graph G = (V ; E) and a positive integer k. Parameter: k Question: Does there exist a subset V 0 V of size at most k such ⊆ that for every edge( u; v) E either u V 0 or v V 0? 2 2 2 Path Input: A graph G = (V ; E) and a positive integer k. Parameter: k Question: Does there exist a path P in G of length at least k? 4 Kernelization: A Method for Everyone Informally: A kernelization algorithm is a polynomial-time transformation that transforms any given parameterized instance to an equivalent instance of the same problem, with size and parameter bounded by a function of the parameter. 5 Kernel: Formally Formally: A kernelization algorithm, or in short, a kernel for a parameterized problem L Σ∗ N is an algorithm that given ⊆ × (x; k) Σ∗ N, outputs in p( x + k) time a pair( x 0; k0) Σ∗ N such that 2 × j j 2 × • (x; k) L (x 0; k0) L , 2 () 2 • x 0 ; k0 f (k), j j ≤ where f is an arbitrary computable function, and p a polynomial. -
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. -
Matroid Theory Release 9.4
Sage 9.4 Reference Manual: Matroid Theory Release 9.4 The Sage Development Team Aug 24, 2021 CONTENTS 1 Basics 1 2 Built-in families and individual matroids 77 3 Concrete implementations 97 4 Abstract matroid classes 149 5 Advanced functionality 161 6 Internals 173 7 Indices and Tables 197 Python Module Index 199 Index 201 i ii CHAPTER ONE BASICS 1.1 Matroid construction 1.1.1 Theory Matroids are combinatorial structures that capture the abstract properties of (linear/algebraic/...) dependence. For- mally, a matroid is a pair M = (E; I) of a finite set E, the groundset, and a collection of subsets I, the independent sets, subject to the following axioms: • I contains the empty set • If X is a set in I, then each subset of X is in I • If two subsets X, Y are in I, and jXj > jY j, then there exists x 2 X − Y such that Y + fxg is in I. See the Wikipedia article on matroids for more theory and examples. Matroids can be obtained from many types of mathematical structures, and Sage supports a number of them. There are two main entry points to Sage’s matroid functionality. The object matroids. contains a number of con- structors for well-known matroids. The function Matroid() allows you to define your own matroids from a variety of sources. We briefly introduce both below; follow the links for more comprehensive documentation. Each matroid object in Sage comes with a number of built-in operations. An overview can be found in the documen- tation of the abstract matroid class. -
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. -
Covering Vectors by Spaces: Regular Matroids∗
Covering vectors by spaces: Regular matroids∗ Fedor V. Fomin1, Petr A. Golovach1, Daniel Lokshtanov1, and Saket Saurabh1,2 1 Department of Informatics, University of Bergen, Norway, {fedor.fomin,petr.golovach,daniello}@ii.uib.no 2 Institute of Mathematical Sciences, Chennai, India, [email protected] Abstract We consider the problem of covering a set of vectors of a given finite dimensional linear space (vector space) by a subspace generated by a set of vectors of minimum size. Specifically, we study the Space Cover problem, where we are given a matrix M and a subset of its columns T ; the task is to find a minimum set F of columns of M disjoint with T such that that the linear span of F contains all vectors of T . This is a fundamental problem arising in different domains, such as coding theory, machine learning, and graph algorithms. We give a parameterized algorithm with running time 2O(k) · ||M||O(1) solving this problem in the case when M is a totally unimodular matrix over rationals, where k is the size of F . In other words, we show that the problem is fixed-parameter tractable parameterized by the rank of the covering subspace. The algorithm is “asymptotically optimal” for the following reasons. Choice of matrices: Vector matroids corresponding to totally unimodular matrices over ration- als are exactly the regular matroids. It is known that for matrices corresponding to a more general class of matroids, namely, binary matroids, the problem becomes W[1]-hard being parameterized by k. Choice of the parameter: The problem is NP-hard even if |T | = 3 on matrix-representations of a subclass of regular matroids, namely cographic matroids. -
Lecture 13 — February, 1 2014 1 Overview 2 Matroids
Advanced Graph Algorithms Jan-Apr 2014 Lecture 13 | February, 1 2014 Lecturer: Saket Saurabh Scribe: Sanjukta Roy 1 Overview In this lecture we learn what is Matroid, the connection between greedy algorithms and matroids. We also look at some examples of Matroids e.g., Linear Matroids, Graphic Matroids etc. 2 Matroids 2.1 A Greedy Approach Let G = (V, E) be a connected undirected graph and let w : E ! R≥0 be a weight function on the edges. For MWST Kruskal's so-called greedy algorithm works. Consider Maximum Weight Matching problem. 1 a b 3 3 d c 4 Application of the greedy algorithm gives (d,c) and (a,b). However, (d,c) and (a,b) do not form a matching of maximum weight. It is obviously not true that such a greedy approach would lead to an optimal solution for any combinatorial optimization problem. It turns out that the structures for which the greedy algorithm does lead to an optimal solution, are the matroids. 2.2 Matroids Definition 1. A pair M = (E, I), where E is a ground set and I is a family of subsets (called independent sets) of E, is a matroid if it satisfies the following conditions: (I1) φ 2 IorI = ø: 1 (I2) If A0 ⊆ A and A 2 I then A0 2 I. (I3) If A, B 2 I and jAj < jBj, then 9e 2 (B n A) such that A [feg 2 I: The axiom (I2) is also called the hereditary property and a pair M = (E, I) satisfying (I1) and (I2) is called hereditary family or set-family. -
Partial Fields and Matroid Representation
ADVANCES IN APPLIED MATHEMATICS 17, 184]208Ž. 1996 ARTICLE NO. 0010 Partial Fields and Matroid Representation Charles Semple and Geoff Whittle Department of Mathematics, Victoria Uni¨ersity, PO Box 600 Wellington, New Zealand Received May 12, 1995 A partial field P is an algebraic structure that behaves very much like a field except that addition is a partial binary operation, that is, for some a, b g P, a q b may not be defined. We develop a theory of matroid representation over partial fields. It is shown that many important classes of matroids arise as the class of matroids representable over a partial field. The matroids representable over a partial field are closed under standard matroid operations such as the taking of minors, duals, direct sums, and 2-sums. Homomorphisms of partial fields are defined. It is shown that if w: P12ª P is a non-trivial partial-field homomorphism, then every matroid representable over P12is representable over P . The connec- tion with Dowling group geometries is examined. It is shown that if G is a finite abelian group, and r ) 2, then there exists a partial field over which the rank-r Dowling group geometry is representable if and only if G has at most one element of order 2, that is, if G is a group in which the identity has at most two square roots. Q 1996 Academic Press, Inc. 1. INTRODUCTION It follows from a classical result of Tuttewx 19 that a matroid is repre- sentable over GFŽ.2 and some field of characteristic other than 2 if and only if it can be represented over the rationals by the columns of a totally unimodular matrix, that is, by a matrix over the rationals all of whose non-zero subdeterminants are inÄ4 1, y1 . -
Arxiv:1910.05689V1 [Math.CO]
ON GRAPHIC ELEMENTARY LIFTS OF GRAPHIC MATROIDS GANESH MUNDHE1, Y. M. BORSE2, AND K. V. DALVI3 Abstract. Zaslavsky introduced the concept of lifted-graphic matroid. For binary matroids, a binary elementary lift can be defined in terms of the splitting operation. In this paper, we give a method to get a forbidden-minor characterization for the class of graphic matroids whose all lifted-graphic matroids are also graphic using the splitting operation. Keywords: Elementary lifts; splitting; binary matroids; minors; graphic; lifted-graphic Mathematics Subject Classification: 05B35; 05C50; 05C83 1. Introduction For undefined notions and terminology, we refer to Oxley [13]. A matroid M is quotient of a matroid N if there is a matroid Q such that, for some X ⊂ E(Q), N = Q\X and M = Q/X. If |X| = 1, then M is an elementary quotient of N. A matroid N is a lift of M if M is a quotient of N. If M is an elementary quotient of N, then N is an elementary lift of M. A matroid N is a lifted-graphic matroid if there is a matroid Q with E(Q)= E(N) ∪ e such that Q\e = N and Q/e is graphic. The concept of lifted-graphic matroid was introduced by Zaslavsky [19]. Lifted-graphic matroids play an important role in the matroid minors project of Geelen, Gerards and Whittle [9, 10]. Lifted- graphic matroids are studied in [4, 5, 6, 7, 19]. This class is minor-closed. In [5], it is proved that there exist infinitely many pairwise non-isomorphic excluded minors for the class of lifted-graphic matroids. -
Matroid Optimization and Algorithms Robert E. Bixby and William H
Matroid Optimization and Algorithms Robert E. Bixby and William H. Cunningham June, 1990 TR90-15 MATROID OPTIMIZATION AND ALGORITHMS by Robert E. Bixby Rice University and William H. Cunningham Carleton University Contents 1. Introduction 2. Matroid Optimization 3. Applications of Matroid Intersection 4. Submodular Functions and Polymatroids 5. Submodular Flows and other General Models 6. Matroid Connectivity Algorithms 7. Recognition of Representability 8. Matroid Flows and Linear Programming 1 1. INTRODUCTION This chapter considers matroid theory from a constructive and algorithmic viewpoint. A substantial part of the developments in this direction have been motivated by opti mization. Matroid theory has led to a unification of fundamental ideas of combinatorial optimization as well as to the solution of significant open problems in the subject. In addition to its influence on this larger subject, matroid optimization is itself a beautiful part of matroid theory. The most basic optimizational property of matroids is that for any subset every max imal independent set contained in it is maximum. Alternatively, a trivial algorithm max imizes any {O, 1 }-valued weight function over the independent sets. Most of matroid op timization consists of attempts to solve successive generalizations of this problem. In one direction it is generalized to the problem of finding a largest common independent set of two matroids: the matroid intersection problem. This problem includes the matching problem for bipartite graphs, and several other combinatorial problems. In Edmonds' solu tion of it and the equivalent matroid partition problem, he introduced the notions of good characterization (intimately related to the NP class of problems) and matroid (oracle) algorithm.