Strict Gammoids Research Articles

Oriented cobicircular matroids are GSP

Colourings and flows are well-known dual notions in graph theory. In turn, the definition of flows in graphs naturally extends to flows in oriented matroids. So, the colour-flow duality gives a generalization of Hadwiger's conjecture about graph colourings, to a conjecture about coflows of oriented matroids. The first non-trivial case of Hadwiger's conjecture for oriented matroids reads as follows. If O is an M(K4)-minor free oriented matroid, then O has a nowhere-zero 3-coflow, i.e., it is 3-colourable in the sense of Hochstättler-Nešetřil. The class of generalized series parallel (GSP) oriented matroids is a class of 3-colourable oriented matroids with no M(K4)-minor. So far, the only technique towards proving that all orientations of a class C of M(K4)-minor free matroids are GSP (and thus 3-colourable), has been to show that every matroid in C has a positive coline. Towards proving Hadwiger's conjecture for the class of gammoids, Goddyn, Hochstättler, and Neudauer conjectured that every gammoid has a positive coline. In this work we disprove this conjecture by showing that there are infinitely many strict gammoids that do not have positive colines. We conclude by proposing a simpler technique for showing that certain oriented matroids are GSP. In particular, we recover that oriented lattice path matroids are GSP, and we show that oriented cobicircular matroids are GSP.

Presentations of transversal valuated matroids

Given $d$ row vectors of $n$ tropical numbers, $d<n$, the tropical Stiefel map constructs a version of their row space, whose Pl\"ucker coordinates are tropical determinants. We explicitly describe the fibers of this map. From the viewpoint of matroid theory, the tropical Stiefel map defines a generalization of transversal matroids in the valuated context, and our results are the valuated generalizations of theorems of Brualdi and Dinolt, Mason and others on the set of all set families that present a given transversal matroid. We show that a connected valuated matroid is transversal if and only if all of its connected initial matroids are. The duals of our results describe complete stable intersections via valuated strict gammoids.

Linear representation of transversal matroids and gammoids parameterized by rank

Given a bipartite graph G=(U⊎V,E), a linear representation of the transversal matroid associated with G on the ground set U, can be constructed in randomized polynomial time. In fact one can get a linear representation deterministically in time 2O(m2n), where m=|U| and n=|V|, by looping through all the choices made in the randomized algorithm. Other important matroids for which one can obtain linear representation deterministically in time similar to the one for transversal matroids include gammoids and strict gammoids. Strict gammoids are duals of transversal matroids and gammoids are restrictions of strict gammoids. We give faster deterministic algorithms to construct linear representations of transversal matroids, gammoids and strict gammoids. All our algorithms run in time (mr)mO(1), where m is the cardinality of the ground set and r is the rank of the matroid. In the language of parameterized complexity, we give an XP algorithm for finding linear representations of transversal matroids, gammoids and strict gammoids parameterized by the rank of the given matroid.

On Finding Some New Excluded Minors for Gammoids

The class of gammoids is the smallest class of matroids closed under duality and minors that contains the class of transversal matroids. Ingleton showed that the class of gammoids does not allow for a finite characterization in terms of excluded minors [Ingleton, A., “Transversal Matroids and Related Structures,” Springer Netherlands, Dordrecht, 1977 pp. 117–131], and Mayhew showed that every gammoid may be extended to a matroid that is an excluded minor for the class of gammoids [Mayhew, D., The antichain of excluded minors for the class of gammoids is maximal, ArXiv e-prints (2016)]. The properties of the antichain of excluded minors has yet to be examined thoroughly. The natural approach to this would be to start with some non-gammoid and then examine its minors for the minimal minors that are not gammoids. Although it is comparatively easy to check whether a given matroid is either a strict gammoid or a transversal matroid, we still have to deal with minors that are neither strict gammoids nor transversal matroids. We introduce an easily implemented and automated sufficient condition that a given matroid is not a gammoid, which greatly reduces the handwork needed in this investigation.

Infinite Gammoids: Minors and Duality

This sequel to Afzali Borujeni et. al. (2015) considers minors and duals of infinite gammoids. We prove that the class of gammoids defined by digraphs not containing a certain type of substructure, called an outgoing comb, is minor-closed. Also, we prove that finite-rank minors of gammoids are gammoids. Furthermore, the topological gammoids of Carmesin (2014) are proved to coincide, as matroids, with the finitary gammoids. A corollary is that topological gammoids are minor-closed.It is a well-known fact that the dual of any finite strict gammoid is a transversal matroid. The class of strict gammoids defined by digraphs not containing alternating combs, introduced in Afzali Borujeni et. al. (2015), contains examples which are not dual to any transversal matroid. However, we describe the duals of matroids in this class as a natural extension of transversal matroids. While finite gammoids are closed under duality, we construct a strict gammoid that is not dual to any gammoid.

Finitary and cofinitary gammoids

A gammoid is a matroid defined using linkability of vertex sets in a (possibly infinite) digraph. Related types of matroids are strict gammoids and transversal matroids, three aspects of which will be considered as follows.First, we investigate the interaction between matroid properties of strict gammoids and transversal matroids and graphic properties of the defining graphs. In particular, we characterize cofinitary strict gammoids and cofinitary transversal matroids among, respectively, strict gammoids and transversal matroids in terms of the defining graphs.The set of finite circuits of a matroid defines the finitarization matroid on the same ground set. A matroid is nearly finitary if every base can be extended to a base of the finitarization matroid by adding finitely many elements. Aigner-Horev et al. (2011) [6] raised the question whether the number of such additions is bounded for a fixed nearly finitary matroid. We answer this question positively in the classes of strict gammoids and transversal matroids.Piff and Welsh (1970) proved the classical result that a finite strict gammoid/transversal matroid is representable over any large enough field. In this direction, we prove that finitary strict gammoids and transversal matroids are representable over some field, and hence the cofinitary counterparts are thin sums representable, a new notion of representability introduced by Bruhn and Diestel (2011).

Gammoids, Pseudomodularity and Flatness Degree

We introduce the concept of flatness degree for matroids, as a generalization of submodularity. This represents weaker variations of the concept of flatness which characterize strict gammoids for finite matroids. We prove that having flatness degree 3, which is the smallest non-trivial flatness degree, implies pseudomodularity on the lattice of flats of the matroid. We show however an example of a gammoid for which the converse is not true. We also show examples of gammoids with each possible flatness degree. All of this examples show that pseudomodular gammoids are not necessarily strict.

Infinite Gammoids

Finite strict gammoids, introduced in the early 1970's, are matroids defined via finite digraphs equipped with some set of sinks: a set of vertices is independent if it admits a linkage to these sinks. In particular, an independent set is maximal (i.e. a base) precisely if it is linkable onto the sinks.In the infinite setting, this characterization of the maximal independent sets need not hold. We identify a type of substructure as the unique obstruction. This allows us to prove that the sets linkable onto the sinks form the bases of a (possibly non-finitary) matroid if and only if this substructure does not occur.

Transversal and Cotransversal Matroids via their Representations

It is known that the duals of transversal matroids are precisely the strict gammoids. We show that, by representing these two families of matroids geometrically, one obtains a simple proof of their duality.

Full transversal matroids, strict gammoids, and the matroid components problem

We provide two algorithms for finding dependence graphs both in a full transversal matroid and in its dual, a strict gammoid. The first algorithm is based on directed paths in the directed graph associated with a strict gammoid; its complexity is O(|L|(|V−L|+|E|)), where L is the link-set of the gammoid. The second algorithm is based on a special property of Gaussian elimination in a matrix of indeterminates representing a full transversal matroid; it complexity is o(m2n), where m is the rank of the matroid and n the cardinality of the underlying set. We provide an algorithm for listing all bases in, and calculating the Whitney and Tutte polynomials for, a full transversal matroid or a strict gammoid. The complexity of this algorithm is 0(N(n−m) (|E| + m 2)), where N is the number of bases.

Undirected strict gammoids

Undirected strict gammoids

Gammoids and transversal matroids

The two main results of this paper identify the “strict gammoids” of Mason [7] with duals of transvesal matroids, and gammoids in general with contractions of transversal matroids. Both theorems derive from a fundamental construction which we also use, inter alia, to establish a duality between the graph theorems of Menger and König.

Strict Gammoids and Rank Functions

Strict Gammoids and Rank Functions