Problems in Generic Combinatorial Rigidity: Sparsity, Sliders, and Emergence of Components
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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. -
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. -
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 . -
Matroid Representation of Clique Complexes∗
Matroid representation of clique complexes¤ x Kenji Kashiwabaray Yoshio Okamotoz Takeaki Uno{ Abstract In this paper, we approach the quality of a greedy algorithm for the maximum weighted clique problem from the viewpoint of matroid theory. More precisely, we consider the clique complex of a graph (the collection of all cliques of the graph) which is also called a flag complex, and investigate the minimum number k such that the clique complex of a given graph can be represented as the intersection of k matroids. This number k can be regarded as a measure of “how complex a graph is with respect to the maximum weighted clique problem” since a greedy algorithm is a k-approximation algorithm for this problem. For any k > 0, we characterize graphs whose clique complexes can be represented as the intersection of k matroids. As a consequence, we can see that the class of clique complexes is the same as the class of the intersections of partition matroids. Moreover, we determine how many matroids are necessary and sufficient for the representation of all graphs with n vertices. This number turns out to be n - 1. Other related investigations are also given. Keywords: Abstract simplicial complex, Clique complex, Flag complex, Independence system, Matroid intersec- tion, Partition matroid 1 Introduction An independence system is a family of subsets of a nonempty finite set such that all subsets of a member of the family are also members of the family. A lot of combinatorial optimization problems can be seen as optimization problems on the corresponding independence systems. For example, in the minimum cost spanning tree problem, we want to find a maximal set with minimum total weight in the collection of all forests of a given graph, which is an independence system. -
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 Panolan3, Saket Saurabh4, and Meirav Zehavi5 1 University of Bergen, Bergen, Norway [email protected] 2 The Institute of Mathematical Sciences, HBNI, Chennai, India [email protected] 3 University of Bergen, Bergen, Norway [email protected] 4 University of Bergen, Bergen, Norway and The Institute of Mathematical Sciences, HBNI, Chennai, India [email protected] 5 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. -
Spectral Conditions for Graph Rigidity in the Euclidean Plane
Spectral conditions for graph rigidity in the Euclidean plane Sebastian M. Cioab˘a∗ and Xiaofeng Guy January 20, 2020 Abstract Rigidity is the property of a structure that does not flex. It is well studied in discrete geometry and mechanics and has applications in material science, engineering and biological sciences. A bar-and-joint framework is a pair (G; p) of graph G together with a map p of the vertices of G into the Euclidean d-space. We view the edges of (G; p) as bars and the vertices as universal joints. The vertices can move continuously as long as the distances between pairs of adjacent vertices are preserved. The framework is rigid if any such motion preserves the distances between all pairs of vertices. In 1970, Laman obtained a combina- torial characterization of rigid graphs in the Euclidean plane. In 1982, Lov´aszand Yemini discovered a new characterization and proved that every 6-connected graph is rigid in the Euclidean plane. Consequently, if Fiedler's algebraic connectivity is at least 6, then G is rigid. In this paper, we show that if G has minimum degree δ ≥ 6 and algebraic connectivity 1 greater than 2 + δ−1 , then G is rigid. We prove a more general result giving a necessary spectral condition for packing k edge-disjoint spanning rigid subgraphs. The same condition implies that a graph contains k edge-disjoint 2-connected spanning subgraphs. This result extends previous spectral conditions for packing edge-disjoint spanning trees. MSC: 05C50, 05C70 Key words: eigenvalue, algebraic connectivity, rigid graph, connectivity 1 Introduction In this paper, we consider finite undirected simple graphs. -
Matroids Graphs Give Isomorphic Matroid Structure?
EECS 495: Combinatorial Optimization Lecture 7 Matroid Representation, Matroid Optimization Reading: Schrijver, Chapters 39 and 40 graph and S = E. A set F ⊆ E is indepen- dent if it is acyclic. Food for thought: can two non-isomorphic Matroids graphs give isomorphic matroid structure? Recap Representation Def: A matroid M = (S; I) is a finite ground Def: For a field F , a matroid M is repre- set S together with a collection of indepen- sentable over F if it is isomorphic to a linear dent sets I ⊆ 2S satisfying: matroid with matrix A and linear indepen- dence taken over F . • downward closed: if I 2 I and J ⊆ I, 2 then J 2 I, and Example: Is uniform matroid U4 binary? Need: matrix A with entries in f0; 1g s.t. no • exchange property: if I;J 2 I and jJj > column is the zero vector, no two rows sum jIj, then there exists an element z 2 J nI to zero over GF(2), any three rows sum to s.t. I [ fzg 2 I. GF(2). Def: A basis is a maximal independent set. • if so, can assume A is 2×4 with columns The cardinality of a basis is the rank of the 1/2 being (0; 1) and (1; 0) and remaining matroid. two vectors with entries in 0; 1 neither all k Def: Uniform matroids Un are given by jSj = zero. n, I = fI ⊆ S : jIj ≤ kg. • only three such non-zero vectors, so can't Def: Linear matroids: Let F be a field, A 2 have all pairs indep. -
Some Inequalities for the Tutte Polynomial
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector European Journal of Combinatorics 32 (2011) 422–433 Contents lists available at ScienceDirect European Journal of Combinatorics journal homepage: www.elsevier.com/locate/ejc Some inequalities for the Tutte polynomial Laura E. Chávez-Lomelí a, Criel Merino b,1, Steven D. Noble c, Marcelino Ramírez-Ibáñez b a Universidad Autónoma Metropolitana, Unidad Azcapotzalco, Avenida San Pablo 180, colonia Reynosa Tamaulipas, Delegación Azcapotzalco, Mexico b Instituto de Matemáticas, Universidad Nacional Autónoma de México, Area de la Investigación Científica, Circuito Exterior, C.U. Coyoacán 04510, México D.F., Mexico c Department of Mathematical Sciences, Brunel University, Kingston Lane, Uxbridge UB8 3PH, UK article info a b s t r a c t Article history: We prove that the Tutte polynomial of a coloopless paving matroid Received 22 February 2010 is convex along the portion of the line x C y D p lying in the Accepted 6 September 2010 positive quadrant. Every coloopless paving matroid is in the class Available online 29 December 2010 of matroids which contain two disjoint bases or whose ground set is the union of two bases. For this latter class we give a proof that TM .a; a/ ≤ maxfTM .2a; 0/; TM .0; 2a/g for a ≥ 2. We conjecture that TM .1; 1/ ≤ maxfTM .2; 0/; TM .0; 2/g for the same class of matroids. We also prove this conjecture for some families of graphs and matroids. ' 2010 Elsevier Ltd. All rights reserved. 1. Introduction The Tutte polynomial is a two-variable polynomial which can be defined for a graph G or more generally a matroid M.