Aspects of Connectivity with Matroid Constraints in Graphs Quentin Fortier
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Math Preliminaries
Preliminaries We establish here a few notational conventions used throughout the text. Arithmetic with ∞ We shall sometimes use the symbols “∞” and “−∞” in simple arithmetic expressions involving real numbers. The interpretation given to such ex- pressions is the usual, natural one; for example, for all real numbers x, we have −∞ < x < ∞, x + ∞ = ∞, x − ∞ = −∞, ∞ + ∞ = ∞, and (−∞) + (−∞) = −∞. Some such expressions have no sensible interpreta- tion (e.g., ∞ − ∞). Logarithms and exponentials We denote by log x the natural logarithm of x. The logarithm of x to the base b is denoted logb x. We denote by ex the usual exponential function, where e ≈ 2.71828 is the base of the natural logarithm. We may also write exp[x] instead of ex. Sets and relations We use the symbol ∅ to denote the empty set. For two sets A, B, we use the notation A ⊆ B to mean that A is a subset of B (with A possibly equal to B), and the notation A ( B to mean that A is a proper subset of B (i.e., A ⊆ B but A 6= B); further, A ∪ B denotes the union of A and B, A ∩ B the intersection of A and B, and A \ B the set of all elements of A that are not in B. For sets S1,...,Sn, we denote by S1 × · · · × Sn the Cartesian product xiv Preliminaries xv of S1,...,Sn, that is, the set of all n-tuples (a1, . , an), where ai ∈ Si for i = 1, . , n. We use the notation S×n to denote the Cartesian product of n copies of a set S, and for x ∈ S, we denote by x×n the element of S×n consisting of n copies of x. -
Math 7410 Graph Theory
Math 7410 Graph Theory Bogdan Oporowski Department of Mathematics Louisiana State University April 14, 2021 Definition of a graph Definition 1.1 A graph G is a triple (V,E, I) where ◮ V (or V (G)) is a finite set whose elements are called vertices; ◮ E (or E(G)) is a finite set disjoint from V whose elements are called edges; and ◮ I, called the incidence relation, is a subset of V E in which each edge is × in relation with exactly one or two vertices. v2 e1 v1 Example 1.2 ◮ V = v1, v2, v3, v4 e5 { } e2 e4 ◮ E = e1,e2,e3,e4,e5,e6,e7 { } e6 ◮ I = (v1,e1), (v1,e4), (v1,e5), (v1,e6), { v4 (v2,e1), (v2,e2), (v3,e2), (v3,e3), (v3,e5), e7 v3 e3 (v3,e6), (v4,e3), (v4,e4), (v4,e7) } Simple graphs Definition 1.3 ◮ Edges incident with just one vertex are loops. ◮ Edges incident with the same pair of vertices are parallel. ◮ Graphs with no parallel edges and no loops are called simple. v2 e1 v1 e5 e2 e4 e6 v4 e7 v3 e3 Edges of a simple graph can be described as v e1 v 2 1 two-element subsets of the vertex set. Example 1.4 e5 e2 e4 E = v1, v2 , v2, v3 , v3, v4 , e6 {{ } { } { } v1, v4 , v1, v3 . v4 { } { }} v e7 3 e3 Note 1.5 Graph Terminology Definition 1.6 ◮ The graph G is empty if V = , and is trivial if E = . ∅ ∅ ◮ The cardinality of the vertex-set of a graph G is called the order of G and denoted G . -
Sets, Functions
Sets 1 Sets Informally: A set is a collection of (mathematical) objects, with the collection treated as a single mathematical object. Examples: • real numbers, • complex numbers, C • integers, • All students in our class Defining Sets Sets can be defined directly: e.g. {1,2,4,8,16,32,…}, {CSC1130,CSC2110,…} Order, number of occurence are not important. e.g. {A,B,C} = {C,B,A} = {A,A,B,C,B} A set can be an element of another set. {1,{2},{3,{4}}} Defining Sets by Predicates The set of elements, x, in A such that P(x) is true. {}x APx| ( ) The set of prime numbers: Commonly Used Sets • N = {0, 1, 2, 3, …}, the set of natural numbers • Z = {…, -2, -1, 0, 1, 2, …}, the set of integers • Z+ = {1, 2, 3, …}, the set of positive integers • Q = {p/q | p Z, q Z, and q ≠ 0}, the set of rational numbers • R, the set of real numbers Special Sets • Empty Set (null set): a set that has no elements, denoted by ф or {}. • Example: The set of all positive integers that are greater than their squares is an empty set. • Singleton set: a set with one element • Compare: ф and {ф} – Ф: an empty set. Think of this as an empty folder – {ф}: a set with one element. The element is an empty set. Think of this as an folder with an empty folder in it. Venn Diagrams • Represent sets graphically • The universal set U, which contains all the objects under consideration, is represented by a rectangle. -
Even Factors, Jump Systems, and Discrete Convexity
Even Factors, Jump Systems, and Discrete Convexity Yusuke KOBAYASHI¤ Kenjiro TAKAZAWAy May, 2007 Abstract A jump system, which is a set of integer lattice points with an exchange property, is an extended concept of a matroid. Some combinatorial structures such as the degree sequences of the matchings in an undirected graph are known to form a jump system. On the other hand, the maximum even factor problem is a generalization of the maximum matching problem into digraphs. When the given digraph has a certain property called odd- cycle-symmetry, this problem is polynomially solvable. The main result of this paper is that the degree sequences of all even factors in a digraph form a jump system if and only if the digraph is odd-cycle-symmetric. Furthermore, as a gen- eralization, we show that the weighted even factors induce M-convex (M-concave) functions on jump systems. These results suggest that even factors are a natural generalization of matchings and the assumption of odd-cycle-symmetry of digraphs is essential. 1 Introduction In the study of combinatorial optimization, extensions of matroids are introduced as abstract con- cepts including many combinatorial objects. A number of optimization problems on matroidal structures can be solved in polynomial time. One of the extensions of matroids is a jump system of Bouchet and Cunningham [2]. A jump system is a set of integer lattice points with an exchange property (to be described in Section 2.1); see also [18, 22]. It is a generalization of a matroid [4], a delta-matroid [1, 3, 8], and a base polyhedron of an integral polymatroid (or a submodular sys- tem) [14]. -
Matroid Theory
MATROID THEORY HAYLEY HILLMAN 1 2 HAYLEY HILLMAN Contents 1. Introduction to Matroids 3 1.1. Basic Graph Theory 3 1.2. Basic Linear Algebra 4 2. Bases 5 2.1. An Example in Linear Algebra 6 2.2. An Example in Graph Theory 6 3. Rank Function 8 3.1. The Rank Function in Graph Theory 9 3.2. The Rank Function in Linear Algebra 11 4. Independent Sets 14 4.1. Independent Sets in Graph Theory 14 4.2. Independent Sets in Linear Algebra 17 5. Cycles 21 5.1. Cycles in Graph Theory 22 5.2. Cycles in Linear Algebra 24 6. Vertex-Edge Incidence Matrix 25 References 27 MATROID THEORY 3 1. Introduction to Matroids A matroid is a structure that generalizes the properties of indepen- dence. Relevant applications are found in graph theory and linear algebra. There are several ways to define a matroid, each relate to the concept of independence. This paper will focus on the the definitions of a matroid in terms of bases, the rank function, independent sets and cycles. Throughout this paper, we observe how both graphs and matrices can be viewed as matroids. Then we translate graph theory to linear algebra, and vice versa, using the language of matroids to facilitate our discussion. Many proofs for the properties of each definition of a matroid have been omitted from this paper, but you may find complete proofs in Oxley[2], Whitney[3], and Wilson[4]. The four definitions of a matroid introduced in this paper are equiv- alent to each other. -
The Structure of the 3-Separations of 3-Connected Matroids
THE STRUCTURE OF THE 3{SEPARATIONS OF 3{CONNECTED MATROIDS JAMES OXLEY, CHARLES SEMPLE, AND GEOFF WHITTLE Abstract. Tutte defined a k{separation of a matroid M to be a partition (A; B) of the ground set of M such that |A|; |B|≥k and r(A)+r(B) − r(M) <k. If, for all m<n, the matroid M has no m{separations, then M is n{connected. Earlier, Whitney showed that (A; B) is a 1{separation of M if and only if A is a union of 2{connected components of M.WhenM is 2{connected, Cunningham and Edmonds gave a tree decomposition of M that displays all of its 2{separations. When M is 3{connected, this paper describes a tree decomposition of M that displays, up to a certain natural equivalence, all non-trivial 3{ separations of M. 1. Introduction One of Tutte’s many important contributions to matroid theory was the introduction of the general theory of separations and connectivity [10] de- fined in the abstract. The structure of the 1–separations in a matroid is elementary. They induce a partition of the ground set which in turn induces a decomposition of the matroid into 2–connected components [11]. Cun- ningham and Edmonds [1] considered the structure of 2–separations in a matroid. They showed that a 2–connected matroid M can be decomposed into a set of 3–connected matroids with the property that M can be built from these 3–connected matroids via a canonical operation known as 2–sum. Moreover, there is a labelled tree that gives a precise description of the way that M is built from the 3–connected pieces. -
A Combinatorial Abstraction of the Shortest Path Problem and Its Relationship to Greedoids
A Combinatorial Abstraction of the Shortest Path Problem and its Relationship to Greedoids by E. Andrew Boyd Technical Report 88-7, May 1988 Abstract A natural generalization of the shortest path problem to arbitrary set systems is presented that captures a number of interesting problems, in cluding the usual graph-theoretic shortest path problem and the problem of finding a minimum weight set on a matroid. Necessary and sufficient conditions for the solution of this problem by the greedy algorithm are then investigated. In particular, it is noted that it is necessary but not sufficient for the underlying combinatorial structure to be a greedoid, and three ex tremely diverse collections of sufficient conditions taken from the greedoid literature are presented. 0.1 Introduction Two fundamental problems in the theory of combinatorial optimization are the shortest path problem and the problem of finding a minimum weight set on a matroid. It has long been recognized that both of these problems are solvable by a greedy algorithm - the shortest path problem by Dijk stra's algorithm [Dijkstra 1959] and the matroid problem by "the" greedy algorithm [Edmonds 1971]. Because these two problems are so fundamental and have such similar solution procedures it is natural to ask if they have a common generalization. The answer to this question not only provides insight into what structural properties make the greedy algorithm work but expands the class of combinatorial optimization problems known to be effi ciently solvable. The present work is related to the broader question of recognizing gen eral conditions under which a greedy algorithm can be used to solve a given combinatorial optimization problem. -
Matroids and Tropical Geometry
matroids unimodality and log concavity strategy: geometric models tropical models directions Matroids and Tropical Geometry Federico Ardila San Francisco State University (San Francisco, California) Mathematical Sciences Research Institute (Berkeley, California) Universidad de Los Andes (Bogotá, Colombia) Introductory Workshop: Geometric and Topological Combinatorics MSRI, September 5, 2017 (3,-2,0) Who is here? • This is the Introductory Workshop. • Focus on accessibility for grad students and junior faculty. • # (questions by students + postdocs) # (questions by others) • ≥ matroids unimodality and log concavity strategy: geometric models tropical models directions Preface. Thank you, organizers! • This is the Introductory Workshop. • Focus on accessibility for grad students and junior faculty. • # (questions by students + postdocs) # (questions by others) • ≥ matroids unimodality and log concavity strategy: geometric models tropical models directions Preface. Thank you, organizers! • Who is here? • matroids unimodality and log concavity strategy: geometric models tropical models directions Preface. Thank you, organizers! • Who is here? • This is the Introductory Workshop. • Focus on accessibility for grad students and junior faculty. • # (questions by students + postdocs) # (questions by others) • ≥ matroids unimodality and log concavity strategy: geometric models tropical models directions Summary. Matroids are everywhere. • Many matroid sequences are (conj.) unimodal, log-concave. • Geometry helps matroids. • Tropical geometry helps matroids and needs matroids. • (If time) Some new constructions and results. • Joint with Carly Klivans (06), Graham Denham+June Huh (17). Properties: (B1) B = /0 6 (B2) If A;B B and a A B, 2 2 − then there exists b B A 2 − such that (A a) b B. − [ 2 Definition. A set E and a collection B of subsets of E are a matroid if they satisfies properties (B1) and (B2). -
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Arxiv:1403.0920V3 [Math.CO] 1 Mar 2019
Matroids, delta-matroids and embedded graphs Carolyn Chuna, Iain Moffattb, Steven D. Noblec,, Ralf Rueckriemend,1 aMathematics Department, United States Naval Academy, Chauvenet Hall, 572C Holloway Road, Annapolis, Maryland 21402-5002, United States of America bDepartment of Mathematics, Royal Holloway University of London, Egham, Surrey, TW20 0EX, United Kingdom cDepartment of Mathematics, Brunel University, Uxbridge, Middlesex, UB8 3PH, United Kingdom d Aschaffenburger Strasse 23, 10779, Berlin Abstract Matroid theory is often thought of as a generalization of graph theory. In this paper we propose an analogous correspondence between embedded graphs and delta-matroids. We show that delta-matroids arise as the natural extension of graphic matroids to the setting of embedded graphs. We show that various basic ribbon graph operations and concepts have delta-matroid analogues, and illus- trate how the connections between embedded graphs and delta-matroids can be exploited. Also, in direct analogy with the fact that the Tutte polynomial is matroidal, we show that several polynomials of embedded graphs from the liter- ature, including the Las Vergnas, Bollab´as-Riordanand Krushkal polynomials, are in fact delta-matroidal. Keywords: matroid, delta-matroid, ribbon graph, quasi-tree, partial dual, topological graph polynomial 2010 MSC: 05B35, 05C10, 05C31, 05C83 1. Overview Matroid theory is often thought of as a generalization of graph theory. Many results in graph theory turn out to be special cases of results in matroid theory. This is beneficial -
Matroids You Have Known
26 MATHEMATICS MAGAZINE Matroids You Have Known DAVID L. NEEL Seattle University Seattle, Washington 98122 [email protected] NANCY ANN NEUDAUER Pacific University Forest Grove, Oregon 97116 nancy@pacificu.edu Anyone who has worked with matroids has come away with the conviction that matroids are one of the richest and most useful ideas of our day. —Gian Carlo Rota [10] Why matroids? Have you noticed hidden connections between seemingly unrelated mathematical ideas? Strange that finding roots of polynomials can tell us important things about how to solve certain ordinary differential equations, or that computing a determinant would have anything to do with finding solutions to a linear system of equations. But this is one of the charming features of mathematics—that disparate objects share similar traits. Properties like independence appear in many contexts. Do you find independence everywhere you look? In 1933, three Harvard Junior Fellows unified this recurring theme in mathematics by defining a new mathematical object that they dubbed matroid [4]. Matroids are everywhere, if only we knew how to look. What led those junior-fellows to matroids? The same thing that will lead us: Ma- troids arise from shared behaviors of vector spaces and graphs. We explore this natural motivation for the matroid through two examples and consider how properties of in- dependence surface. We first consider the two matroids arising from these examples, and later introduce three more that are probably less familiar. Delving deeper, we can find matroids in arrangements of hyperplanes, configurations of points, and geometric lattices, if your tastes run in that direction. -
Invitation to Matroid Theory
Invitation to Matroid Theory Gregory Henselman-Petrusek January 21, 2021 Abstract This text is a companion to the short course Invitation to matroid theory taught in the Univer- sity of Oxford Centre for TDA in January 2021. It was first written for algebraic topologists, but should be suitable for all audiences; no background in matroid theory is assumed! 1 How to use this text Matroid theory is a beautiful, powerful, and accessible. Many important subjects can be understood and even researched with just a handful of concepts and definitions from matroid theory. Even so, getting use out of matroids often depends on knowing the right keywords for an internet search, and finding the right keyword depends on two things: (i) a precise knowledge of the core terms/definitions of matroid theory, and (ii) a birds eye view of the network of relationships that interconnect some of the main ideas in the field. These are the focus of this text; it falls far short of a comprehensive overview, but provides a strong start to (i) and enough of (ii) to build an interesting (and fun!) network of ideas including some of the most important concepts in matroid theory. The proofs of most results can be found in standard textbooks (e.g. Oxley, Matroid theory). Exercises are chosen to maximize the fraction intellectual reward time required to solve Therefore, if an exercise takes more than a few minutes to solve, it is fine to look up answers in a textbook! Struggling is not prerequisite to learning, at this level. 2 References The following make no attempt at completeness. -
Ear Decomposition of Factor-Critical Graphs and Number of Maximum Matchings ∗
Ear Decomposition of Factor-critical Graphs and Number of Maximum Matchings ¤ Yan Liu, Shenlin Zhang Department of Mathematics, South China Normal University, Guangzhou, 510631, P.R. China Abstract A connected graph G is said to be factor-critical if G ¡ v has a perfect matching for every vertex v of G. Lov´asz proved that every factor-critical graph has an ear decomposition. In this paper, the ear decomposition of the factor-critical graphs G satisfying that G ¡ v has a unique perfect matching for any vertex v of G with degree at least 3 is characterized. From this, the number of maximum matchings of factor-critical graphs with the special ear decomposition is obtained. Key words maximum matching; factor-critical graph; ear decomposition 1 Introduction and terminology First, we give some notation and definitions. For details, see [1] and [2]. Let G be a simple graph. An edge subset M ⊆ E(G) is a matching of G if no two edges in M are incident with a common vertex. A matching M of G is a perfect matching if every vertex of G is incident with an edge in M. A matching M of G is a maximum matching if jM 0j · jMj for any matching M 0 of G. Let v be a vertex of G. The degree of v in G is denoted by dG(v) and ±(G) =minfdG(v) j v 2 V (G)g. Let P = u1u2 ¢ ¢ ¢ uk be a path and 1 · s · t · k. Then usus+1 ¢ ¢ ¢ ut is said to be a subpath of P , denoted by P (us; ut).