Graphic Matroids Research Articles

Intersecting set systems and graphic matroids

Two simple proofs are given to an earlier partial result about an extremal set theoretic conjecture of Chung, Frankl, Graham, Shearer and Faudree, Schelp, Sós, respectively. The statement is slightly strengthened within a matroid theoretic framework. The first proof relies on results from matroid theory, while the second is based on an explicit construction providing an elementary proof.

On coefficients of the Tutte polynomial

This paper characterizes, for each i and j, the matroids that are minor-minimal among connected matroids M with b ij ( M) > 0, where t( M) = Σb ij ( M) x i y j is the Tutte polynomial of M. One consequence of this characterization for a connected matroid M is that b 11( M) > 0 if and only if the two-wheel is a minor of M. Similar results are obtained for other small values of i and j. A generalization of these results leads to new combinatorial proofs which strengthen known results on the coefficients. These results imply that if M is simple and representable over GF( q), then there are coefficients of its Tutte polynomial which count the flats of M of each rank that are projective spaces. Similarly, for a simple graphic matroid M( G), there are coefficients that count the number of cliques of each size contained in G.

A 2-Isomorphism Theorem for Hypergraphs

One can associate a polymatroid with a hypergraph that naturally generalises the cycle matroid of a graph. Whitney's 2-isomorphism theorem characterises when two graphs have isomorphic cycle matroids. In this paper Whitney's theorem is generalised to hypergraphs and polymatroids by characterising when two hypergraphs have isomorphic associated polymatroids.

Unavoidable Minors of Large 3-Connected Matroids

This paper proves that, for every integernexceeding two, there is a numberN(n) such that every 3-connected matroid with at leastN(n) elements has a minor that is isomorphic to one of the following matroids: an (n+2)-point line or its dual, the cycle or cocycle matroid ofK3,n, the cycle matroid of a wheel withnspokes, a whirl of rankn, or ann-spike. A matroid is of the last type if it has ranknand consists ofnthree-point lines through a common point such that, for allkin {1,2,…,n−1}, the union of every set ofkof these lines has rankk+1.

Binary matroids without prisms, prism duals, and cubes

In this paper we determine completely the class of binary matroids with no minors isomorphic to the cycle matroid of the prism graph M *( K 5 e), its dual M( K 5 e), and the binary affine cube AG(3,2).

Unavoidable Minors of Large 3-Connected Binary Matroids

We show that, for every integer n greater than two, there is a number N such that every 3-connected binary matroid with at least N elements has a minor that is isomorphic to the cycle matroid of K 3, n , its dual, the cycle matroid of the wheel with n spokes, or the vector matroid of the binary matrix ( I n | J n − I n ), where J n is the n × n matrix of all ones.

Self-dual graphs

We consider the three forms of self-duality that can be exhibited by a planar graph G, map self-duality, graph self-duality and matroid self-duality. We show how these concepts are related with each other and with the connectivity of G. We use the geometry of self-dual polyhedra together with the structure of the cycle matroid to construct all self-dual graphs.

Base exchange properties of graphic matroids

New base exchange properties of binary and graphic matroids are derived. The graphic matroids within the class of 4-connected binary matroids are characterized by base exchange properties. Some progress with the characterization of arbitrary graphic matroids is made. Characterizing various types of matroids by base exchange properties is e.g. important in invariant theory.

Constrained matroidal bottleneck problems

Algorithms for the constrained bottleneck matroid base problem and the constrained bottleneck independent set problem are presented. The algorithms have improved complexities upon the best known general algorithm. Sufficient conditions are obtained when the proposed algorithms have linear complexity and are therefore the best possible ones. Several specific classes of matroidal problems are considered, including problems on partition, graphic and uniform matroids, special variants of the assignment and scheduling problems. The proposed approach is based on shrinking iteratively the dimensions of auxiliary problems. It is demonstrated that the approach cannot be extended beyond the class of matroidal problems, and the assumption of matroidal structure is not only sufficient but also necessary for the applicability of the approach.

On Tutt's Characterization of graphic matroids—a graphic proof

AbstractIn this paper we present a relatively simple proof of Tutt's characterization of graphic matroids. The proof uses the notion of ‘signed graph’ and it is ‘graphic’ in the sense that it can be presented almost entirely by drawing (signed) graphs. © 1995 John Wiley & Sons, Inc.

Algebraic graph theory without orientation

Let G be an undirected graph with vertices {v 1,v 2,…,>;v ⋎} and edges { e 1, e 2, …, e ϵ }. Let M be the ⋎ × ϵ matrix whose ijth entry is 1 if e j is a link incident with v i , 2 if e j is a loop at v i , and 0 otherwise. The matrix obtained by orienting the edges of a loopless graph G (i.e., changing one of the 1's to a − 1 in each column of M) has been studied extensively in the literature. The purpose of this paper is to explore the substructures of G and the vector spaces associated with the matrix M without imposing such an orientation. We describe explicitly bases for the kernel and range of the linear transformation from R ϵ to R ⋎ defined by M. Our main results are determinantal formulas, using the unoriented Laplacian matrix MM t , to count certain spanning substructures of G. These formulas may be viewed as generalizations of the matrix tree theorem. The point of view adopted in this paper also gives rise to a matroid structure on the edges of G analogous to the cycle matroid and its dual. In this setting, the analogue of a spanning forest can have components with one odd cycle, and the analogue of an edge cut has the property that its removal creates a new bipartite component.

Abstract and Generic Rigidity in the Plane

We consider the concept of abstract two-dimensional rigidity and provide necessary and sufficient conditions for a matroid to be an abstract rigidity matroid of a complete graph. This characterization is a natural extension of the characterization of graphic matroids due to Graver or Sachs. We also give an example of an abstract rigidity matroid which is not infinitesimal.

Elementary strong maps of graphic matroids, II

In Part I a graph theoretical tool was presented to construct elementary strong maps of graphic matroids, and it was conjectured that every such map arises in this way. An example of Jeff Kahn disproves this conjecture but leads to a generalization of the original strong map construction.

Recognizing Polymatroids Associated with Hypergraphs

Two natural classes of polymatroids can be associated with hypergraphs: the so-called Boolean and hypergraphic polymatroids. Boolean polymatroids carry virtually all the structure of hypergraphs; hypergraphic polymatroids generalize graphic matroids. This paper considers algorithmic problems associated with recognizing members of these classes. Let k be a fixed positive integer and assume that the k-polymatroid ρ is presented via a rank oracle. We present an algorithm that determines in polynomial time whether ρ is Boolean, and if it is, finds the hypergraph. We also give an algorithm that decides in polynomial time whether ρ is the hypergraphic polymatroid associated with a given hypergraph. Other structure-theoretic results are also given.

Dilworth truncations and k-induced matroids

It is known that given a matroid Q on a set S and a function Ø : E → 2 S , the subsets I of E such that | I'|⩽ r Q (Ø( I′)) − k for all non-empty subsets I′ of I are the independent sets of a matroid on E. This matroid is said to be k-induced on Q by Ø. The aim of this paper is to prove that if k and n are nonnegative integers and M is a matroid ( k + n)-induced on Q, then M is k-induced on the nth Dilworth truncation of Q. As a consequence, if M is 1-induced on a free matroid, then M is 0-induced on a graphic matroid Another consequence, a generalization of a well-known result of Mason and Brylawski, is that for any integer k, if Q is representable over a field F and M is k-induced on Q, then M is representable over an extension of F. The converses of these statements do not hold in general.

Ordered h-hypertrees

In this paper a special class of h-uniform hypergraphs, called μ-ordered h-hypertrees relatively to a total order μ on the vertex set is introduced. Seven other equivalent definitions of such hypertrees are proposed involving the notions of μ-ordered path and μ-ordered cycle. It is also proved that for a fixed order μ the sets of edges of μ-ordered h-hypertrees with the same vertex set is the set of bases of a graphic matroid and every h-hypertree can be recognized in polynomial time.

Fractional matroid matchings

This paper introduces a linear relaxation of the matroid matching problem, called the fractional matroid matching problem. When the matroid matching problem is in fact a matroid intersection problem, the fractional matroid matching polytope and the matroid matching polytope coincide. When the matroid matching problem is in fact a matching problem in a graph, the fractional matroid matching polytope and the classic fractional matching polytope of the graph coincide. Thus, the fractional matroid matching polytope may properly contain the matroid matching polytope. The fractional matroid matching polytope is a lattice polyhedron and, although its extreme points are not all integral, they are half-integral. This paper establishes strong relationships between extreme points of the fractional matroid matching polytope and those of graphic fractional matching polyhedra. Despite these strong relationships, adding the rank 1 inequalities does not define the matroid matching polytope. In fact, even the matching polytope of a graphic matroid is not generally described by adding these inequalities.

A minimal 3-connectedness result for matroids

In this paper we prove the following theorem: Let M be a 3-connected matroid other than the cycle matroid of a wheel of rank greater than three. Let C be a circuit of M. If every deletion of a pair of elements of C disconnects M, then every pair of elements of C is in a triad. A result of Oxley and the author, which characterizes the duals of Sylvester matroids, is an immediate consequence of this theorem. It also follows that if such a matroid M is graphic, then C is a 3-edge cycle.

Biased graphs. II. The three matroids

A biased graph Ω consists of a graph Γ and a class B of circles (simple, closed paths) in Γ, called balanced circles, such that no theta subgraph contains exactly two balanced circles. The bias matroid G(Ω) is a finitary matroid on the edge set E of Ω whose circuits are the balanced circles and the minimal connected edge sets of cyclomatic number two containing no balanced circle. We prove that these circuits define a matroid and we establish cryptomorphic definitions and other properties. Another finitary matroid on E, the lift matroid L(Ω), and its one-point extension the complete lift matroid L 0(Ω) , are obtained from the abstract matroid lift construction applied to the graphic matroid G( Γ) and the class B . The circuits of L(Ω) are the balanced circles and the minimal edge sets of cyclomatic number two (not necessarily connected) containing no balanced circle. We develop cryptomorphisms and other properties of L 0(Ω) and L(Ω). There is no completely general construction rule, besides the bias and lift constructions, which assigns to each biased graph a matroid intermediate (in the sense of independent sets) between G and L and which respects subgraphs. G(Ω) has an infinitary analog for infinite graphs generalizing Klee's infinitary bicircular matroid and the Bean-Higgs infinitary graphic matroid. Whether L(Ω) has an infinitary analog is unclear.

Minor characterization of undirected branching greedoids—a short proof

In 1959 Tutte [5] gave a minor characterization of graphic matroids. Within the framework of greedoids, a natural analogue of the cycle matroid in graphs is the branching greedoid. Schmidt has shown that, similar to Tutte's result, branching greedoids can be characterized by forbidden minors. Here we give a simpler proof of this theorem.