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Finitary and cofinitary gammoids
Discrete Applied Mathematics (
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Finitary and cofinitary gammoids Seyed Hadi Afzali Borujeni ∗ , Hiu-Fai Law, Malte Müller Fachbereich Mathematik, Bundestrasse 55, 20146 Hamburg, Germany
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Article history: Received 24 September 2014 Received in revised form 14 July 2015 Accepted 29 July 2015 Available online xxxx Keywords: Infinite matroid Gammoid Transversal matroid Finitary
abstract 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). © 2015 Published by Elsevier B.V.
1. Introduction Transversal matroids and gammoids are defined via the concepts of matchability and linkability in, respectively, bipartite graphs and digraphs (see Edmonds and Fulkerson [14], Mason [19] and Perfect [24]). In case the defining graphs are finite, a matroid structure can be endowed on the linkable sets or matchable sets. In an infinite graph, these concepts can be studied via the notion of infinite matroids. Classically, infinite matroids are finitary, in the sense that a set is independent if and only if all its finite subsets are independent. However, with this definition, the dual of a matroid, for example a uniform matroid, need not be a matroid. In response to this, there has been a variety of proposals to define infinite matroids allowing for duality (for more on the history, see Higgs [16], Oxley [21,22] and Bruhn et al. [11]). Recently, simple axiomatizations of infinite matroids that extend transparently the rich finite matroid theory were given by Bruhn et al. [11]. Since then, there has been an ongoing project ([4,3] and [12]) which focuses on infinite gammoids and transversal matroids. Here we contribute in three ways: by giving graph theoretic characterizations of cofinitary strict gammoids and transversal matroids, proving the existence of certain bounds on extensions of bases when these types of matroids are nearly finitary, as well as showing that any strict gammoid or transversal matroid that is finitary or cofinitary is thin sums representable.
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Corresponding author. E-mail address: [email protected] (S.H. Afzali Borujeni).
http://dx.doi.org/10.1016/j.dam.2015.07.030 0166-218X/© 2015 Published by Elsevier B.V.
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Independent sets of a transversal matroid are the matchable subsets of a fixed, say left, vertex class in a bipartite graph; while those of a strict gammoid are the sets linkable to a fixed set of sinks of a digraph. As in [4], an ordered pair of a digraph and a fixed set of sinks will be called a dimaze, where these sinks are called exits. As the bipartite graph and the dimaze influence the corresponding matroids strongly, it is of interest to study how graph properties, which are often easy to visualize, force certain matroid properties. For example, consider finitary transversal matroids. A simple compactness proof (see for example [20]) shows that a bipartite graph, with every vertex in the left vertex class of finite degree, defines a finitary transversal matroid. This condition on the bipartite graph in fact characterizes among transversal matroids the finitary ones, namely, a transversal matroid is finitary if and only if it can be defined via a bipartite graph all of whose vertices in the ground set have finite degree (Proposition 3.2). In [12], Carmesin gave a similar characterization for finitary strict gammoids (Theorem 3.3) in terms of forbidden families of digraphs in a defining dimaze. We first focus on the classes of cofinitary transversal matroids and cofinitary strict gammoids. It is not hard to see that if we have a strict gammoid defined on a dimaze, then the set of vertices linkable to a fixed exit constitutes a cocircuit, as this is a minimal set meeting every base. Thus, any dimaze defining a cofinitary strict gammoid cannot have, for example, an infinite directed ray arriving at an exit. By analysing more precisely structures that lead to infinite cocircuits, as well as adapting a conversion between matchings and linkages of Ingleton and Piff [17], we prove characterizations of cofinitary transversal matroids (Theorem 3.8) and cofinitary strict gammoids (Theorem 3.10). Our second topic concerns nearly finitary matroids, which were introduced in [6] to encompass a class larger than finitary matroids, in which one has a matroid union theorem. The finitarization of a matroid is defined by declaring a set independent as soon as all its finite subsets are independent in the given matroid. A matroid is nearly finitary if we can only add finitely many elements to any base while preserving independence in the finitarization. In general, the number of elements added may vary across bases (see Section 4 for an example). An open problem is whether there is a bound on the number of element additions over all bases of a given nearly finitary matroid. We prove the existence of such a bound for nearly finitary strict gammoids and nearly finitary transversal matroids in Propositions 4.2 and 4.8. Traditionally, representable matroids are defined by linear independence in a family of vectors of a vector space and are necessarily finitary. It was proved by Lindström [18], and Piff and Welsh [25], that any given finite gammoid is representable over any large enough field. Infinite gammoids need not be representable anymore as not every infinite gammoid is finitary. To provide a means to represent (possibly non-finitary) infinite matroids, Bruhn and Diestel [10] introduced the notion of thin sums representability, which was later proved to generalize representability by the first author and Bowler [2]. Our investigation along this line leads us to a transversal matroid and a strict gammoid that are not thin sums representable. However, we show that if a strict gammoid or a transversal matroid is finitary or cofinitary, then it is thin sums representable (Proposition 5.1 and Corollary 5.2). 2. Preliminaries We collect definitions and notations. For those not found here, we refer the reader to [11] and [23] for matroid theory, and [13] for graph theory. Analogous to finite matroids, infinite matroids can be axiomatized in different terms, including independent sets, circuits, bases, rank function, and closure operator [11]. The main difference from the finite case is that all of the infinite axiomatizations contain a central axiom which demands for the existence of certain maximal sets. In this paper, we find it convenient to work with the independence axioms. Given a set E and I ⊆ 2E , let Imax denote the maximal elements of I with respect to set inclusion. For sets I and {x}, I + x stands for I ∪ {x}. A matroid M is a pair (E , I), which satisfies the following: (I1) ∅ ∈ I; (I2) if I ⊆ I ′ and I ′ ∈ I then I ∈ I; (I3) for every I ′ ∈ Imax and I ∈ I \ Imax , there is an x ∈ I ′ \ I such that I + x ∈ I; and (IM) whenever I ∈ I and I ⊆ X ⊆ E, the set {I ′ ∈ I : I ⊆ I ′ ⊆ X } has a maximal element. We call E the ground set of M. A subset of E is independent if it is in I, dependent otherwise. As usual, we will identify M with its set of independent sets. Bases are the maximal independent sets and circuits are the minimal dependent sets. The dual of M is a matroid M ∗ on the same ground set whose bases are the complements of the bases of M. Given a set X ⊆ E, a deletion minor or a restriction M \ X = M |(E \ X ) is a matroid on E \ X whose independent sets are I ∩ 2E \X . The contraction of M to X is defined as (M ∗ |X )∗ and denoted by M .X = M /(E \ X ). Given a set system M = (E , I), let I(M fin ) := {I ⊆ E : all finite subsets of Iare in I(M )}. Then the finitarization M fin of M is defined as (E , I(M fin )). It follows that any minimal dependent set of M fin is a finite minimal dependent set of M. We say that M is finitary if and only if M = M fin . We now turn to one of our main objects which are set systems defined via systems of paths in digraphs equipped with a distinguished set of vertices. Let D = (V , E ) be a digraph and B0 ⊆ V a set of sinks. Call the pair (D, B0 ) a dimaze, which is an abbreviation for directed maze, and B0 the (set of) exits. A linkage is a set of vertex disjoint paths ending in B0 . A subset of V is linkable if there is a linkage whose set of initial vertices includes the given set. The pair of V and the set of linkable subsets of V is denoted by ML (D, B0 ). If ML (D, B0 ) is a matroid, it is called a strict gammoid and (D, B0 ) is called a presentation
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of ML (D, B0 ). A gammoid is a matroid restriction of a strict gammoid. In general, ML (D, B0 ) satisfies (I1), (I2) and (I3) but not necessarily (IM) [4]. A dimaze (D′ , B′0 ) is a subdivision of (D, B0 ) if it can be obtained from (D, B0 ) as follows. We first add an extra vertex b0 and the edges {(b, b0 ) : b ∈ B0 } to D. Then the edges of this resulting digraph are subdivided to define a digraph D′′ . Set B′0 as the in-neighbourhood of b0 in D′′ and D′ as D′′ − b0 . Note that this defaults to the usual notion of subdivision if B0 = ∅. The following dimazes play specific roles in the current work. An undirected ray R is a graph with an infinite vertex set X = {xi : i ≥ 1} and edge set {xi xi+1 : i ≥ 1}. Let Y = {yi : i ≥ 1} be a set disjoint from X . (1) RI is obtained from R by orienting (xi+1 , xi ) for each i ≥ 1 and declaring x1 as the only exit; (2) C O is obtained from R by orienting (xi , xi+1 ) for each i ≥ 1, adding the edges (xi , yi ) for each i ≥ 2 and declaring the sinks of the digraph to be the exits; (3) C A is obtained from R by orienting (xi+1 , xi ), (xi+1 , xi+2 ) for each odd i ≥ 1 and declaring the sinks of the digraph to be the exits; and (4) F ∞ is obtained from a star with countably infinitely many leaves by directing the edges towards the leaves and declaring the leaves to be the exits. Any subdivision of RI , C O , C A and F ∞ is called, respectively, an incoming ray, an outgoing comb, an alternating comb and a linking fan. The subdivided ray in any comb is called the spine and the paths from the spine to the exits are the spikes. A dimaze (D, B0 ) contains another dimaze (D′ , B′0 ) if D′ is a subdigraph of D and B′0 ⊆ B0 . For a set H of dimazes, we say that a dimaze (D, B0 ) is H -free if (D, B0 ) does not contain any subdivision of a dimaze in H . A (strict) gammoid is called H -free if it admits a H -free presentation. Let G be a bipartite graph with a fixed vertex bipartition (V , W ). A subset I of V is matchable if there is a matching m covering I. An m-alternating walk is a path or a ray such that consecutive edges alternate between m and non m-edges in G. The pair of V and all its matchable subsets is denoted by MT (G). When MT (G) is a matroid, it is called a transversal matroid, and G is a presentation of MT (G). Generally, MT (G) satisfies (I1), (I2) and (I3) but not necessarily (IM). To see an example which violates (IM), consider a complete bipartite graph which has as V the real numbers, and W the natural numbers. Then clearly every countable subset of V is matchable, but there is no maximal such set, hence (IM) is violated. Whenever there is a maximal matchable set in MT (G), having some matching m0 , it is easy to prove [9] that deleting vertices in W \ V (m0 ) does not change MT (G). In such cases, we may assume that W is covered by a matching of some maximally matchable set. 3. Characterization results In this section, we characterize cofinitary strict gammoids and transversal matroids in graph theoretic terms. For completeness, we collect the analogous results for finitary ones. A left locally finite bipartite graph G = (V , W ), i.e. a bipartite graph such that the vertices in V have finite degree, defines a finitary transversal matroid (see [20]). Starting with any bipartite graph G = (V , W ), let L ⊆ V be the set of vertices of infinite degree and L′ := {v ′ : v ∈ L} disjoint from V ∪ W . Construct the bipartite graph G′ = (V , W ∪ L′ ) by deleting all the edges incident with vertices in L and adding the edges {vv ′ : v ∈ L}. Lemma 3.1. Given a bipartite graph G, with the above definition of G′ , we have MT (G′ ) = MT (G)fin . Proof. Let L ⊆ V be the vertices that have infinite degree in G. Then both G − L and G′ are left locally finite, so that MT (G − L) and MT (G′ ) are finitary. To prove the lemma, it suffices to show that the set of circuits of MT (G′ ) and that of MT (G)fin both equal that of MT (G − L). First note that any circuit C of MT (G′ ) cannot contain vertices in L, as every such vertex has a private neighbour. Hence, C is a circuit of MT (G − L). On the other hand, recall that any circuit of MT (G)fin is a finite circuit of MT (G), which cannot contain vertices in L as any such vertex has infinitely many available neighbours. Hence, the set of circuits of MT (G)fin is the same as that of MT (G − L), thereby completing the proof. If G defines a finitary transversal matroid, then since G′ is a left locally finite graph, we have the following characterization. Proposition 3.2. A transversal matroid is finitary if and only if it admits a left locally finite presentation. Next, as there is no infinite circuit in a finitary strict gammoid, none of its presentation contains any subdivision of C O or F . Indeed, any vertex v together with all the vertices in B0 to which v can be linked form a circuit. In particular, the initial vertex of the spine of an outgoing comb or the centre of a linking fan gives rise to an infinite circuit. The converse is also true. In [12], it is proved that the finitarization of a strict gammoid is obtained by replacing linkages by topological linkages in the definition of independent sets. A topological linkage is a collection of disjoint topological paths, each of which is defined as being the spine of an outgoing comb, a path ending in the centre of a linking fan, or a path ending in B0 . In particular, if a dimaze is {C O , F ∞ }-free, then topological linkages coincide with linkages, leading to the following theorem. ∞
Theorem 3.3 ([12]). Given a dimaze (D, B0 ), independent sets of ML (D, B0 )fin are precisely topologically linkable subsets of V . Consequently, a dimaze defines a finitary strict gammoid if and only if it is {C O , F ∞ }-free.
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Let us turn to cofinitary transversal matroids and strict gammoids. First we recall a construction in [17] that converts a digraph presentation of a finite gammoid to a bipartite graph presentation of its dual transversal matroid. In the following we adjust this construction for our purposes. Given a bipartite graph G = (V , W ) and a matching m0 onto W , we call the pair (G, m0 ) a bimaze, which is an abbreviation for bipartite maze. Definition 3.4. Given a dimaze (D, B0 ), define a bipartite graph D⋆B0 , with bipartition (V , (V \ B0 )⋆ ), where (V \ B0 )⋆ := {v ⋆ : v ∈ V \ B0 } is disjoint from V ; and E (D⋆B0 ) := m0 ∪ {v u⋆ : (u, v) ∈ E (D)}, where m0 := {vv ⋆ : v ∈ V \ B0 }. Call (D, B0 )⋆ := (D⋆B0 , m0 ) the converted bimaze of (D, B0 ). Starting from a dimaze (D, B0 ), we write (V \ B0 )⋆ , m0 and v ⋆ for the corresponding objects in Definition 3.4. The inverse construction is defined as follows: Definition 3.5. Given a bimaze (G, m0 ), where G = (V , W ), define a digraph G⋆m0 such that V (G⋆m0 ) := V and E (G⋆m0 ) := {(v, w) : wv ⋆ ∈ E (G)\ m0 }, where v ⋆ is the vertex in W that is matched by m0 to v ∈ V . Let B0 := V \ V (m0 ). Call (G, m0 )⋆ := (G⋆m0 , B0 ) the converted dimaze of (G, m0 ). Starting from a bimaze (G, m0 ), we write B0 and v ⋆ for the corresponding objects in Definition 3.5 and (V \ B0 )⋆ for W . These constructions are inverse to each other. In particular,
(G, m0 )⋆⋆ = (G, m0 ).
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Given a bimaze (G, m0 ), a set I ⊆ V is m0 -matchable if there is a matching m of I with each connected component of G [m0 ∪ m] finite. The correspondence between linkages in a dimaze and matchings in its converted bimaze is as follows. We remark that the proof of the duality result in [17] relies on the finite counterpart of this correspondence. Lemma 3.6 ([3]). Let (D, B0 ) be a dimaze. Then B is linkable onto B0 in (D, B0 ) if and only if V \ B is m0 -matchable onto (V \ B0 )⋆ in (D, B0 )⋆ . The following is the main tool to prove the two characterization theorems. Theorem 3.7 ([3]). If a dimaze (D, B0 ) is {RI , C A }-free, then ML (D, B0 ) is dual to MT (D⋆B0 ). Similarly, given a bimaze (G, m0 ), if
(G, m0 )⋆ is {RI , C A }-free, then MT (G) is a matroid dual to ML (G, m0 )⋆ .
Recall that any codependent set of a matroid meets every base. Suppose that B is a base and I an independent set such that |B \ I | = |I \ B| < ∞, then one can apply the base exchange axiom [11] to see that I is also a base. Theorem 3.8. Any transversal matroid is cofinitary if and only if it admits a presentation G with a matching m0 of a base such that (G, m0 )⋆ is {RI , C O , C A , F ∞ }-free. Proof. The backward direction follows from Theorems 3.3 and 3.7. Conversely, suppose that M is a cofinitary transversal matroid with a presentation G. Let m0 be a matching of a base V \ B0 of M, then the dimaze (G, m0 )⋆ is RI -free. Suppose not, then there is an incoming ray ending in some b ∈ B0 , which converts to an infinite m0 -alternating walk starting from b in (G, m0 ). This walk gives rise to a matching of (V \ B0 ) + b which contradicts that m0 is a matching of a base of M. For a contradiction, let R be a subdivision of C A in (G, m0 )⋆ . Let I1 be the set of vertices of R with out-degree two, B1 := I1 ∪ (B0 \ V (R)), and v ∈ B0 ∩ V (R). By the construction of R, B1 and B1 + v are both linkable onto B0 . Therefore, by Lemma 3.6 and (1), V \ (B1 + v) is a non-maximal independent set of M. So the complement B1 + v is codependent and contains a cocircuit C of M. Since M is cofinitary, the set C1 := C ∩ I1 is finite. By definition of I1 , C1 is linkable onto some T1 ⊆ B0 \ (B1 + v). Let B2 := C1 ∪ B0 \ T1 . Then |B0 \ B2 | = |B2 \ B0 | < ∞, so that V \ B2 is also a base of M. But this base avoids the cocircuit C . This contradiction shows that (G, m0 )⋆ is C A -free. As (G, m0 )⋆ is {RI , C A }-free, by Theorem 3.7, M ∗ = ML (G, m0 )⋆ . As M ∗ is finitary, (G, m0 )⋆ does not contain any subdivision of C O or F ∞ either. Together with Theorem 3.7, we have the following. Corollary 3.9. Any cofinitary transversal matroid is dual to a strict gammoid. Next, we turn to cofinitary strict gammoids. Theorem 3.10. Any strict gammoid is cofinitary if and only if it admits a {C A , RI }-free presentation such that the in-degree of each vertex is finite.
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Proof. Let M be a strict gammoid. Suppose it has a {C A , RI }-free presentation (D, B0 ) where the in-degree of every vertex is finite. By Theorem 3.7, M ∗ = MT (D⋆B0 ). As the in-degree of D is finite, D⋆B0 is left locally finite. So by Proposition 3.2, MT (D⋆B0 ) is finitary and hence M is cofinitary. Conversely, suppose that M is a cofinitary and admits (D, B0 ) as a presentation. As all the vertices in D linkable to a fixed vertex in B0 form a cocircuit, (D, B0 ) is RI -free. The same reason implies also that every vertex with infinite in-degree cannot be linked to B0 , hence, such a vertex is a loop. Therefore, deleting all the edges incident in such a vertex results in another presentation (D′ , B0 ) of M. For a contradiction, let R be a subdivision of C A in (D′ , B0 ). Let I1 be the set of vertices of R with out-degree two and B1 := I1 ∪ (B0 \ V (R)). Since B1 is a non-maximal independent set, V \ B1 is codependent and contains a cocircuit C of M. Since M is cofinitary, C ∩ B0 is a finite set. Then B0 \ C can be extended by some |C ∩ B0 | vertices of I1 to a base disjoint from C which contradicts the codependence of C . This contradiction shows that (D′ , B0 ) is C A -free. Analogous to Corollary 3.9, applying Theorem 3.7, we have the following. Corollary 3.11. Any cofinitary strict gammoid is dual to a transversal matroid. We remark that neither Corollary 3.9 nor Corollary 3.11 remains true when we drop the cofinitary assumption (see [3]). 4. Nearly finitary gammoids Classical matroid theory usually studies matroids without infinite circuits. On the other hand, recent investigations have led to a growing list of matroids that do contain infinite circuits. Finitarizing a matroid is a process to disregard infinite circuits, thereby relates new infinite matroids to classical ones. Thus, for example (see [2]), the finitarization of a thin sums matroid is a representable matroid, whereas that of a bond matroid of a graph G is the finite bond matroid of G [10]. Nearly finitary matroid is a bridge between finitary and general infinite matroids in the sense that it is ‘only finitely far away’ from a finitary matroid. A matroid M is called nearly finitary [6] if we can delete finitely many points from every base of the finitarization M fin to get a base of M. Equivalently, every base can be extended to a base of the finitarization by adding finitely many points. If the number of deletions in the former, or equivalently, additions in the latter is bounded by an integer k ≥ 0, then M is called k-nearly finitary. Note that in a nearly finitary matroid the number of deletions or additions may depend on the base. For example, in the algebraic cycle matroid of the one-way infinite ladder (for more information, see [11]), a spanning tree without double ray is a base of the matroid as well as its finitarization, whereas two disjoint rays spanning the graph constitute a base, to which an edge can be added without forming any finite cycle. The problem [7] to determine whether a nearly finitary matroid is k-nearly finitary for some k ≥ 0 is open. We will solve this problem for strict gammoids and transversal matroids. For strict gammoids, we are going to use Theorem 3.3 and the following modified version of a result of Halin [15]. We mimic a proof of Andreae [13, Theorem 8.2.5]. Given a path P = x0 · · · xi · · · xj · · · xn , let xi P := xi · · · xn , Pxi := x0 · · · xi , xi Pxj := xi · · · xj and x˚i P := xi+1 · · · xn . For a ray R with an initial vertex, xi R, Rxi , xi Rxj and x˚i R are defined analogously. Given an outgoing comb C and v on its spine R, we write C v for the graph obtained by taking the union of the initial segment Rv of R and the spikes of C that meet Rv . C v is called an initial segment of C . Given two outgoing combs C and C ′ and the initial segments C v and C ′ v ′ , we say that C v is a proper initial segment of C ′ v ′ if Rv is a proper initial segment of R′ v ′ and the set of spikes of C v is a proper subset of that of C ′ v ′ . Lemma 4.1. If for every k ∈ N there is a set of spine-disjoint outgoing combs of size k, then there is an infinite such set. Proof. For any outgoing comb Cin , let Rni be its spine. We construct our infinite set of outgoing combs inductively. In step n ≥ 0, we shall find n spine-disjoint outgoing combs C1n , . . . , Cnn and choose initial segments Cin xni of them. In step n + 1, we +1 n +1 +1 n +1 n +1 n n choose the outgoing combs C1n+1 , . . . , Cnn+ , . . . , xnn+ xi . Then the 1 and x1 1 so that Ci xi is a proper initial segment of Ci graph Ci∗ := n∈N Cin xni is an outgoing comb and (Ci∗ )i∈N is an infinite family of spine-disjoint outgoing combs. We start from the empty set of initial segments of outgoing combs. Suppose that, in step n, C1n , . . . , Cnn and their initial segments C1n xn1 , . . . , Cnn xnn are given, then the construction at step n + 1 is as follows. Abbreviate Cin , Rni and xni to Ci , Ri and xi , respectively. Let C be any set of n2 + 1 spine-disjoint outgoing combs that do not meet any of C1 x1 , . . . , Cn xn . For 1 ≤ i ≤ n, if x˚i Ri meets at most n of the spines of outgoing combs in C , we delete those outgoing combs from C , put Cin+1 := Ci , and choose as xni +1 the first vertex on x˚i Ri where Ci has a spike. Let I := {i ≤ n : Cin+1 is still undefined}, and put m := |I |. Then C still contains at least n2 + 1 − (n − m)n ≥ m2 + 1 outgoing combs. For every i ∈ I, let zi be a vertex on Ri such that xi Ri zi meets exactly m distinct spines of outgoing combs in C . Then Z := i∈I xi Ri zi meets at most m2 spines of +1 the outgoing combs in C . Delete the other outgoing combs from C and choose one of these deleted combs as Cnn+ 1 together 1 with an arbitrary xnn+ +1 on its spine. For each outgoing comb in C , we choose on the spine R a vertex y(R) beyond the vertices of R in Z such that there is a spike at y(R). Let Y := {y(R) : R is the spine of an outgoing comb in C }. Then X := {xi : i ∈ I } clearly cannot be separated from Y in the union H of Z and the paths Ry by fewer than m vertices. Apply Menger’s theorem to find m disjoint X –Y paths
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Pi = xi · · · yi (i ∈ I ) in H. For each i ∈ I, let Ci′ denote the outgoing comb from C whose spine contains yi . Now choose as Cin+1 the outgoing comb obtained by concatenating Ci xi , Pi and yi Ci′ , and put xni +1 := yi . Proposition 4.2. A nearly finitary strict gammoid is k-nearly finitary for some k ≥ 0. Proof. Let M = ML (D, B0 ) be a nearly finitary strict gammoid. By Theorem 3.3, there cannot be infinitely many spinedisjoint outgoing combs, since otherwise the union of the initial vertices of their spines with B0 is independent in M fin , thereby rendering M not nearly finitary. It follows then, by Lemma 4.1, that the size of any set of spine-disjoint outgoing combs is bounded. Similarly, there cannot be infinitely many vertices which are the centre of a linking fan, otherwise, B0 together with these centres form an independent set in M fin . Let k be the sum of the maximum number of spine-disjoint outgoing combs and the number of centres of linking fans. Let B be an independent set in M fin . By Theorem 3.3, there is a topological linkage of B to B0 . In such a linkage, vertices may be the initial vertex of the spine of an outgoing comb or the initial vertex of a path leading to the centre of a linking fan, or a path to B0 . By our estimate B contains at most k vertices of the first two types. Deleting these vertices leaves us with a linkable subset of B. Thus, M is k-nearly finitary. Next, we will prove that any nearly finitary transversal matroid is k-nearly finitary for some k. In fact, our proof depends only on the property that no infinite circuit is contained in a union of finite circuits. Note that a transversal matroid M satisfies this property. Indeed, suppose M contains an infinite circuit C and M = MT (G) for some bipartite graph G. Then G − (V \ C ) is a presentation of the circuit C . By the contrapositive of Proposition 3.2, there is a vertex in C having infinite degree in G. Since such a vertex cannot lie in any finite circuit of M, C is not contained in a union of finite circuits. Lemma 4.3. Let M be a matroid such that no infinite circuit is contained in a union of finite circuits. If M is nearly finitary, then M is k-nearly finitary for some integer k ≥ 0. Proof. Let L be the set of coloops of M fin . Then each element in M \ L is contained in a finite circuit. By the assumption on M, M \ L does not contain any infinite circuit. Given a base B f of M fin , then L ⊆ B f and B f \ L is a base of the finitary matroid M \ L, since B f \ L does not contain any finite circuit of M and addition of any element in M \ B f creates one. Let B1 be a base of M .L. Then B = B1 ∪ (B f \ L) is a base of M. Since M is nearly finitary, B f \ B is finite. On the other hand, B f \ B = L \ B1 . As B f was arbitrary, M is k-nearly finitary for k := |L \ B1 |. Remark 4.4. In fact, if B is a base of M and B f is a base of M fin containing B, then |B f \ B| = k. Indeed, B ∩ L contains a base B1 of M .L, and B \ L can be extended in B f \ L to a base B2 of M \ L. In fact, B2 = B f \ L, so |B f \ B| = |B f \ (B1 ∪ B2 )| = k. In case M is a transversal matroid, the extension of a base of M to one of M fin occurs in the set of vertices of infinite degree. Hence, we have the following. Proposition 4.5. For any integer k ≥ 0 and any bipartite graph G = (V , W ) such that there are at most k vertices of infinite degree in V , MT (G) is a k-nearly finitary matroid. The converse is true for k ≤ 1, that is, a k-nearly finitary transversal matroid admits a presentation with at most k vertices in V of infinite degree. For k = 0, we apply Proposition 3.2. For k = 1, let L be the set of vertices of infinite degree of V in G. We may assume that a base BL of MT (G).L misses exactly one vertex v in L. We form a graph G′ by moving all edges on BL to v , except a matching m of a base of MT (G) extending BL , i.e. we replace each edge e = uw ∈ E (G) − m with u ∈ BL − v by the edge vw . A long case analysis then shows that G′ defines the same transversal matroid as MT (G). We omit the proof. On the other hand, a 2-nearly finitary transversal matroid may only have presentations with infinitely many vertices of infinite degree. Example 4.6. The circuits of the transversal matroid M defined by the bipartite graph in Fig. 1 are {V − x1 , V − x2 , V \ Vi : i ∈ N}, where Vi = {vij : j ∈ N}. For any presentation G = (V , W ) of M, there are infinitely many vertices of infinite degree in V . Proof. It is straightforward to compute the set of circuits. Let X = {x1 , x2 }. Fix a matching m of the base V \ X . Let (D, X ) = (G, m)⋆ . It suffices to prove that in D, there is a vertex of infinite in-degree in each Vi . Note that an (infinite) m-alternating walk in G corresponds to an (infinite) directed path in D. As V − x1 and V − x2 are circuits, there is no incoming ray to X in D. Since V \ Vi is a circuit, there are no two disjoint m-alternating walks from X to Vi in G. Therefore, there are no two disjoint paths from Vi to X in D. By Menger’s theorem, Vi is separated from X by a vertex vi . For any v ∈ Vi , v ′ ∈ Vj where i ̸= j, as V \ {v, v ′ } is matchable, there is a pair of disjoint paths from {v, v ′ } to X . It follows that vi ∈ Vi . For any v ∈ Vi − vi , if there is an edge (v, v ′ ) with v ′ ∈ Vj , then the edge can be extended to a path from v to X avoiding vi , using a path in a linkage from {v ′ , vi } to X , which is a contradiction. Hence, within Vi , there is a path from v to vi . As there is no incoming ray to X , it follows that Vi contains a vertex of infinite in-degree. So having at most k vertices of infinite degree is sufficient but not necessary for a transversal matroid to be k-nearly finitary. However, we can delete some k vertices of a k-nearly finitary transversal matroid to get a finitary minor. To be precise, we have the following.
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Fig. 1. A 2-nearly finitary transversal matroid.
Proposition 4.7. Let M be a matroid such that no infinite circuit is contained in a union of finite circuits, then M is k-nearly finitary if and only if there is a set X of k points such that M \ X is finitary. Proof. Suppose that M is k-nearly finitary. Let L be the set of coloops of M fin . By the assumption on M, M \ L is a finitary matroid. Let B1 be a base of M .L and B2 be a base of M \ L. We claim that X := L \ B1 is the required set. As B1 ∪ B2 ∪ X is a base of M fin , |X | ≤ k. It remains to show that M \ X is finitary. Let C be a circuit of M \ X . As C ⊆ B1 ∪ (C \ L) and B1 is a base of M .L, C ∩ L = ∅. So C is a circuit of the finitary matroid M \ L. Conversely, let X be any set of size at most k such that M \ X is finitary and let Bfin be a base of M fin . As any infinite circuit must meet X , Bfin \ X is independent in M, and we can extend it inside X ∩ Bfin to a base B′ of M. As Bfin is arbitrary and |Bfin \ B′ | ≤ |X | ≤ k, M is k-nearly finitary. Note that for the backward direction we do not need any assumption on M. Finally, Lemma 4.3 and the paragraph before that together with Proposition 4.7 imply the following. Proposition 4.8. Let M be a transversal matroid. The following are equivalent:
• M is nearly finitary. • M is k-nearly finitary, for some integer k ≥ 0. • There exists a set X of k vertices such that M \ X is finitary. A consequence of Theorem 3.3 is that a strict gammoid is k-nearly finitary if and only if each of its presentations contains a total of at most k spine-disjoint outgoing combs and centres of linking fans. In line with our interest in graph theoretic characterization, it would be interesting to answer the following. Problem 4.9. Characterize k-nearly finitary transversal matroids in graph theoretic terms. 5. Representability of gammoids It was proved in [25] that any finite transversal matroid (and hence any finite strict gammoid by duality) is representable over any large enough field. We first use the arguments in [18,19] to sketch a proof that any finitary gammoid is representable. For a function f : E → kA for some field k and set A, let M (f ) denote the vector matroid generated by f , i.e. I(M (f )) = {I ⊆ E : f (I ) is a linearly independent set of vectors over k}. Proposition 5.1. A finitary transversal matroid is representable, and so is a finitary strict gammoid. Proof. We need only to prove that a finitary strict gammoid ML (D, B0 ) is representable, since a finitary transversal matroid is the restriction of some finitary strict gammoid. Let Y = {yij : (i, j) ∈ E (D)} be a set of algebraically independent indeterminates. Let kY be the field of fractions of the formal power series of Y over the rationals. Associate to each edge B (i, j) the weight yij . Define g : V → kY0 such that gv (b) is the sum of the weight of all the directed walks from v to b ∈ B0 , where the weight of any such walk is the product of the weight of its edges.
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Fig. 2. A wild transversal matroid (where the bold edges form a matching of a base) and its strict gammoid dual.
If I is a finite independent set, then there is a linkage from I onto some B ⊆ B0 . By expanding the determinant of the square matrix defined by g with columns in I and rows in B as in [18], we conclude that I is independent in M (g ). Since both ML (D, B0 ) and M (g ) are finitary, this shows that any independent set in ML (D, B0 ) is also independent in M (g ). On the other hand, let C be a circuit in ML (D, B0 ). By Menger’s theorem, there is a separator S of size |C | − 1 separating ′ ′′ ′ C from B0 . Note that for v ∈ C , gv (b) = s∈S gv (s)gs (b) where gv (s) is the sum of weights of walks from v to s ∈ S in D and gs′′ (b) is the sum of weights of walks from s to b in the modified digraph obtained from D by deleting the edges entering S. Since |S | < |C |, there is a nonzero vector (λv : v ∈ C ) such that v∈C λv gv′ (s) = 0 for any s ∈ S. Hence, v∈C λv gv (b) = 0 for all b ∈ B0 . Thus, C is dependent in M (g ). We conclude that ML (D, B0 ) = M (g ). By definition, a representable matroid is finitary. In order to capture non-finitary matroids, a generalization called thin sums representability was proposed in [10]. Given sets E , A and a field k and a function f : E → kA mapping e to fe : A → k, a thin dependence is a function c : E → k such that for each a ∈ A, {e ∈ E : c (e)fe (a) ̸= 0} is finite and e∈E c (e)fe (a) = 0. Let Mts (f ) denote the pair E with the collection of subsets of E each of which does not admit a nonzero thin dependence. In case Mts (f ) is a matroid, it is called a thin sums matroid, and any matroid isomorphic to Mts (f ) for some f : E → kA thin sums representable. It was proved in [2] that a representable matroid is always thin sums representable. While it is not true that Mts (f ) is a matroid for every f , it is the case when f is thin, i.e. {e ∈ E : fe (a) ̸= 0} is finite for each a. In fact, Mts (f ) is cofinitary if f is thin [2]. A matroid is tame if the intersection of a circuit and a cocircuit is always finite, wild otherwise [8]. Since the class of tame thin sums matroids is closed under duality [2], we have the following corollary by applying Corollaries 3.9 and 3.11. Corollary 5.2. A cofinitary transversal matroid is thin sums representable, and so is a cofinitary strict gammoid. It is not true that every transversal matroid or strict gammoid is thin sums representable over some field. We will show that the bipartite graph in Fig. 2 defines a wild transversal matroid that is not thin sums representable over any field. We note that the bipartite graph does not contain any ray. For a proof that any bipartite graph G not containing any ray defines a transversal matroid, see [1, Lemma 3.6.19]. Essentially the proof of (IM) makes use of an infinite version of König’s duality theorem [5] and the fact that the vertices of a matching of G given by this theorem which are in the ground set form a maximal independent set of MT (G) as G does not contain any ray. Lemma 5.3. Let M be a thin sums matroid whose ground set is covered by finitely many circuits. Then M is cofinitary (and hence is not wild). Proof. Let M = Mts (f ) where f : E → kA for some field k, set A and function f . It suffices to show that f is thin, for then we conclude by [2] that Mts (f ) is cofinitary. Observe that for any circuit C , there is a thin dependence c whose support is precisely C . By definition of thin dependence, for each a, {e ∈ E : c (e)fe (a) ̸= 0} = {e ∈ E : fe (a) ̸= 0} ∩ C is finite. It follows that {e ∈ E : fe (a) ̸= 0} is finite, since E is a union of finitely many circuits. Therefore, f is thin as desired. Proposition 5.4. Let G be a bipartite graph such that V (G) = {x, y, ai , bi : i ≥ 1} ∪ {Z , Ai : i ≥ 1} and E (G) = {ai Ai , ai Z , bi Ai , xAi , yAi , xZ , yZ : i ≥ 1} (see Fig. 2). Then MT (G) is a wild transversal matroid not thin sums representable over any field. Moreover, MT∗ (G) is a wild strict gammoid not thin sums representable over any field. Proof. It can be checked that the infinite set {ai : i ≥ 1} + x + y is both a circuit and a cocircuit (and so is {bi : i ≥ 1} + x + y). Hence, MT (G) is wild. Moreover, as these two circuits cover the ground set, by Lemma 5.3, MT (G) is not thin sums representable. By Theorem 3.7, MT∗ (G) is a strict gammoid. Again by Lemma 5.3, MT∗ (G) is not thin sums representable either.
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In view of Propositions 5.1 and 5.2, we propose the following. Conjecture 5.5. A tame gammoid is thin sums representable. Acknowledgements The authors thank the anonymous referees for their valuable comments in a submission of this paper. HFL thanks the support from the Alexander von Humboldt Foundation and the Croucher Foundation. References [1] S.H. Afzali Borujeni, Representability of infinite matroids and the structure of linkages in digraphs (Ph.D. thesis), Universität Hamburg, Germany, 2014, http://ediss.sub.uni-hamburg.de/volltexte/2014/7030/. [2] S.H. Afzali Borujeni, N. Bowler, Thin sums matroids and duality, Adv. Math. 271 (2015) 1–29. [3] S.H. Afzali Borujeni, H.-F. Law, M. Müller, Infinite gammoids: Minors and duality (2014). arXiv:1411.2277v1. [4] S.H. Afzali Borujeni, H.-F. Law, M. Müller, Infinite gammoids, Electron. J. Combin. 22 (2015) # P1.53. [5] R. Aharoni, König’s duality theorem for infinite bipartite graphs, J. Lond. Math. Soc. 29 (1984) 1–12. [6] E. Aigner-Horev, J. Carmesin, J.-O. Fröhlich, Infinite matroid union (2011). arXiv:1111.0602v2. [7] E. Aigner-Horev, J. Carmesin, J.-O. Fröhlich, On the intersection of infinite matroids (2011). arXiv:1111.0606. [8] N. Bowler, J. Carmesin, Matroids with an infinite circuit-cocircuit intersection, J. Combin. Theory Ser. B 107 (2014) 78–91. [9] R.A. Brualdi, E.B. Scrimger, Exchange systems, matchings and transversals, J. Combin. Theory 5 (1968) 244–257. [10] H. Bruhn, R. Diestel, Infinite matroids in graphs, in: Infinite Graph Theory Special Volume of Discrete Math., vol. 311, 2011, pp. 1461–1471. [11] H. Bruhn, R. Diestel, M. Kriesell, R. Pendavingh, P. Wollan, Axioms for infinite matroids, Adv. Math 239 (2013) 18–46. [12] J. Carmesin, Topological infinite gammoids, and a new Menger-type theorem for infinite graphs (2014). arXiv:1404.0151v1. [13] R. Diestel, Graph Theory, fourth ed., Springer, Heidelberg, 2010. [14] J. Edmonds, D.R. Fulkerson, Transversals and matroid partition, J. Res. Natl. Bur. Stand. 69B (1965) 147–153. [15] R. Halin, Über die Maximalzahl fremder unendlicher Wege (in German), Matt. Nachr. 30 (1965) 63–85. [16] D.A. Higgs, Matroids and duality, Colloq. Math. 20 (1969) 215–220. [17] A.W. Ingleton, M.J. Piff, Gammoids and transversal matroids, J. Combin. Theory Ser. B 15 (1973) 51–68. [18] B. Lindström, On the vector representations of induced matroids, Bull. Lond. Math. Soc. 5 (1973) 85–90. [19] J.H. Mason, On a class of matroids arising from paths in graphs, Proc. Lond. Math. Soc. 25 (1972) 55–74. [20] L. Mirsky, H. Perfect, Applications of the notion of independence to problems of combinatorial analysis, J. Combin. Theory 2 (1967) 327–357. [21] J.G. Oxley, Infinite matroids, Proc. Lond. Math. Soc. 37 (1978) 259–272. [22] J.G. Oxley, Some problems in combinatorial geometries (Ph.D. thesis), Oxford University, U.K, 1978. [23] J.G. Oxley, Matroid Theory, Oxford University Press, 1992. [24] H. Perfect, Applications of Menger’s graph theorem, J. Math. Anal. Appl. 22 (1968) 96–111. [25] M.J. Piff, D.J.A. Welsh, On the vector representation of matroids, J. Lond. Math. Soc. 2 (1970) 284–288.
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Discrete Applied Mathematics journal homepage: www.elsevier.com/locate/dam
Finitary and cofinitary gammoids Seyed Hadi Afzali Borujeni ∗ , Hiu-Fai Law, Malte Müller Fachbereich Mathematik, Bundestrasse 55, 20146 Hamburg, Germany
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Article history: Received 24 September 2014 Received in revised form 14 July 2015 Accepted 29 July 2015 Available online xxxx Keywords: Infinite matroid Gammoid Transversal matroid Finitary
abstract 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). © 2015 Published by Elsevier B.V.
1. Introduction Transversal matroids and gammoids are defined via the concepts of matchability and linkability in, respectively, bipartite graphs and digraphs (see Edmonds and Fulkerson [14], Mason [19] and Perfect [24]). In case the defining graphs are finite, a matroid structure can be endowed on the linkable sets or matchable sets. In an infinite graph, these concepts can be studied via the notion of infinite matroids. Classically, infinite matroids are finitary, in the sense that a set is independent if and only if all its finite subsets are independent. However, with this definition, the dual of a matroid, for example a uniform matroid, need not be a matroid. In response to this, there has been a variety of proposals to define infinite matroids allowing for duality (for more on the history, see Higgs [16], Oxley [21,22] and Bruhn et al. [11]). Recently, simple axiomatizations of infinite matroids that extend transparently the rich finite matroid theory were given by Bruhn et al. [11]. Since then, there has been an ongoing project ([4,3] and [12]) which focuses on infinite gammoids and transversal matroids. Here we contribute in three ways: by giving graph theoretic characterizations of cofinitary strict gammoids and transversal matroids, proving the existence of certain bounds on extensions of bases when these types of matroids are nearly finitary, as well as showing that any strict gammoid or transversal matroid that is finitary or cofinitary is thin sums representable.
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Corresponding author. E-mail address: [email protected] (S.H. Afzali Borujeni).
http://dx.doi.org/10.1016/j.dam.2015.07.030 0166-218X/© 2015 Published by Elsevier B.V.
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Independent sets of a transversal matroid are the matchable subsets of a fixed, say left, vertex class in a bipartite graph; while those of a strict gammoid are the sets linkable to a fixed set of sinks of a digraph. As in [4], an ordered pair of a digraph and a fixed set of sinks will be called a dimaze, where these sinks are called exits. As the bipartite graph and the dimaze influence the corresponding matroids strongly, it is of interest to study how graph properties, which are often easy to visualize, force certain matroid properties. For example, consider finitary transversal matroids. A simple compactness proof (see for example [20]) shows that a bipartite graph, with every vertex in the left vertex class of finite degree, defines a finitary transversal matroid. This condition on the bipartite graph in fact characterizes among transversal matroids the finitary ones, namely, a transversal matroid is finitary if and only if it can be defined via a bipartite graph all of whose vertices in the ground set have finite degree (Proposition 3.2). In [12], Carmesin gave a similar characterization for finitary strict gammoids (Theorem 3.3) in terms of forbidden families of digraphs in a defining dimaze. We first focus on the classes of cofinitary transversal matroids and cofinitary strict gammoids. It is not hard to see that if we have a strict gammoid defined on a dimaze, then the set of vertices linkable to a fixed exit constitutes a cocircuit, as this is a minimal set meeting every base. Thus, any dimaze defining a cofinitary strict gammoid cannot have, for example, an infinite directed ray arriving at an exit. By analysing more precisely structures that lead to infinite cocircuits, as well as adapting a conversion between matchings and linkages of Ingleton and Piff [17], we prove characterizations of cofinitary transversal matroids (Theorem 3.8) and cofinitary strict gammoids (Theorem 3.10). Our second topic concerns nearly finitary matroids, which were introduced in [6] to encompass a class larger than finitary matroids, in which one has a matroid union theorem. The finitarization of a matroid is defined by declaring a set independent as soon as all its finite subsets are independent in the given matroid. A matroid is nearly finitary if we can only add finitely many elements to any base while preserving independence in the finitarization. In general, the number of elements added may vary across bases (see Section 4 for an example). An open problem is whether there is a bound on the number of element additions over all bases of a given nearly finitary matroid. We prove the existence of such a bound for nearly finitary strict gammoids and nearly finitary transversal matroids in Propositions 4.2 and 4.8. Traditionally, representable matroids are defined by linear independence in a family of vectors of a vector space and are necessarily finitary. It was proved by Lindström [18], and Piff and Welsh [25], that any given finite gammoid is representable over any large enough field. Infinite gammoids need not be representable anymore as not every infinite gammoid is finitary. To provide a means to represent (possibly non-finitary) infinite matroids, Bruhn and Diestel [10] introduced the notion of thin sums representability, which was later proved to generalize representability by the first author and Bowler [2]. Our investigation along this line leads us to a transversal matroid and a strict gammoid that are not thin sums representable. However, we show that if a strict gammoid or a transversal matroid is finitary or cofinitary, then it is thin sums representable (Proposition 5.1 and Corollary 5.2). 2. Preliminaries We collect definitions and notations. For those not found here, we refer the reader to [11] and [23] for matroid theory, and [13] for graph theory. Analogous to finite matroids, infinite matroids can be axiomatized in different terms, including independent sets, circuits, bases, rank function, and closure operator [11]. The main difference from the finite case is that all of the infinite axiomatizations contain a central axiom which demands for the existence of certain maximal sets. In this paper, we find it convenient to work with the independence axioms. Given a set E and I ⊆ 2E , let Imax denote the maximal elements of I with respect to set inclusion. For sets I and {x}, I + x stands for I ∪ {x}. A matroid M is a pair (E , I), which satisfies the following: (I1) ∅ ∈ I; (I2) if I ⊆ I ′ and I ′ ∈ I then I ∈ I; (I3) for every I ′ ∈ Imax and I ∈ I \ Imax , there is an x ∈ I ′ \ I such that I + x ∈ I; and (IM) whenever I ∈ I and I ⊆ X ⊆ E, the set {I ′ ∈ I : I ⊆ I ′ ⊆ X } has a maximal element. We call E the ground set of M. A subset of E is independent if it is in I, dependent otherwise. As usual, we will identify M with its set of independent sets. Bases are the maximal independent sets and circuits are the minimal dependent sets. The dual of M is a matroid M ∗ on the same ground set whose bases are the complements of the bases of M. Given a set X ⊆ E, a deletion minor or a restriction M \ X = M |(E \ X ) is a matroid on E \ X whose independent sets are I ∩ 2E \X . The contraction of M to X is defined as (M ∗ |X )∗ and denoted by M .X = M /(E \ X ). Given a set system M = (E , I), let I(M fin ) := {I ⊆ E : all finite subsets of Iare in I(M )}. Then the finitarization M fin of M is defined as (E , I(M fin )). It follows that any minimal dependent set of M fin is a finite minimal dependent set of M. We say that M is finitary if and only if M = M fin . We now turn to one of our main objects which are set systems defined via systems of paths in digraphs equipped with a distinguished set of vertices. Let D = (V , E ) be a digraph and B0 ⊆ V a set of sinks. Call the pair (D, B0 ) a dimaze, which is an abbreviation for directed maze, and B0 the (set of) exits. A linkage is a set of vertex disjoint paths ending in B0 . A subset of V is linkable if there is a linkage whose set of initial vertices includes the given set. The pair of V and the set of linkable subsets of V is denoted by ML (D, B0 ). If ML (D, B0 ) is a matroid, it is called a strict gammoid and (D, B0 ) is called a presentation
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of ML (D, B0 ). A gammoid is a matroid restriction of a strict gammoid. In general, ML (D, B0 ) satisfies (I1), (I2) and (I3) but not necessarily (IM) [4]. A dimaze (D′ , B′0 ) is a subdivision of (D, B0 ) if it can be obtained from (D, B0 ) as follows. We first add an extra vertex b0 and the edges {(b, b0 ) : b ∈ B0 } to D. Then the edges of this resulting digraph are subdivided to define a digraph D′′ . Set B′0 as the in-neighbourhood of b0 in D′′ and D′ as D′′ − b0 . Note that this defaults to the usual notion of subdivision if B0 = ∅. The following dimazes play specific roles in the current work. An undirected ray R is a graph with an infinite vertex set X = {xi : i ≥ 1} and edge set {xi xi+1 : i ≥ 1}. Let Y = {yi : i ≥ 1} be a set disjoint from X . (1) RI is obtained from R by orienting (xi+1 , xi ) for each i ≥ 1 and declaring x1 as the only exit; (2) C O is obtained from R by orienting (xi , xi+1 ) for each i ≥ 1, adding the edges (xi , yi ) for each i ≥ 2 and declaring the sinks of the digraph to be the exits; (3) C A is obtained from R by orienting (xi+1 , xi ), (xi+1 , xi+2 ) for each odd i ≥ 1 and declaring the sinks of the digraph to be the exits; and (4) F ∞ is obtained from a star with countably infinitely many leaves by directing the edges towards the leaves and declaring the leaves to be the exits. Any subdivision of RI , C O , C A and F ∞ is called, respectively, an incoming ray, an outgoing comb, an alternating comb and a linking fan. The subdivided ray in any comb is called the spine and the paths from the spine to the exits are the spikes. A dimaze (D, B0 ) contains another dimaze (D′ , B′0 ) if D′ is a subdigraph of D and B′0 ⊆ B0 . For a set H of dimazes, we say that a dimaze (D, B0 ) is H -free if (D, B0 ) does not contain any subdivision of a dimaze in H . A (strict) gammoid is called H -free if it admits a H -free presentation. Let G be a bipartite graph with a fixed vertex bipartition (V , W ). A subset I of V is matchable if there is a matching m covering I. An m-alternating walk is a path or a ray such that consecutive edges alternate between m and non m-edges in G. The pair of V and all its matchable subsets is denoted by MT (G). When MT (G) is a matroid, it is called a transversal matroid, and G is a presentation of MT (G). Generally, MT (G) satisfies (I1), (I2) and (I3) but not necessarily (IM). To see an example which violates (IM), consider a complete bipartite graph which has as V the real numbers, and W the natural numbers. Then clearly every countable subset of V is matchable, but there is no maximal such set, hence (IM) is violated. Whenever there is a maximal matchable set in MT (G), having some matching m0 , it is easy to prove [9] that deleting vertices in W \ V (m0 ) does not change MT (G). In such cases, we may assume that W is covered by a matching of some maximally matchable set. 3. Characterization results In this section, we characterize cofinitary strict gammoids and transversal matroids in graph theoretic terms. For completeness, we collect the analogous results for finitary ones. A left locally finite bipartite graph G = (V , W ), i.e. a bipartite graph such that the vertices in V have finite degree, defines a finitary transversal matroid (see [20]). Starting with any bipartite graph G = (V , W ), let L ⊆ V be the set of vertices of infinite degree and L′ := {v ′ : v ∈ L} disjoint from V ∪ W . Construct the bipartite graph G′ = (V , W ∪ L′ ) by deleting all the edges incident with vertices in L and adding the edges {vv ′ : v ∈ L}. Lemma 3.1. Given a bipartite graph G, with the above definition of G′ , we have MT (G′ ) = MT (G)fin . Proof. Let L ⊆ V be the vertices that have infinite degree in G. Then both G − L and G′ are left locally finite, so that MT (G − L) and MT (G′ ) are finitary. To prove the lemma, it suffices to show that the set of circuits of MT (G′ ) and that of MT (G)fin both equal that of MT (G − L). First note that any circuit C of MT (G′ ) cannot contain vertices in L, as every such vertex has a private neighbour. Hence, C is a circuit of MT (G − L). On the other hand, recall that any circuit of MT (G)fin is a finite circuit of MT (G), which cannot contain vertices in L as any such vertex has infinitely many available neighbours. Hence, the set of circuits of MT (G)fin is the same as that of MT (G − L), thereby completing the proof. If G defines a finitary transversal matroid, then since G′ is a left locally finite graph, we have the following characterization. Proposition 3.2. A transversal matroid is finitary if and only if it admits a left locally finite presentation. Next, as there is no infinite circuit in a finitary strict gammoid, none of its presentation contains any subdivision of C O or F . Indeed, any vertex v together with all the vertices in B0 to which v can be linked form a circuit. In particular, the initial vertex of the spine of an outgoing comb or the centre of a linking fan gives rise to an infinite circuit. The converse is also true. In [12], it is proved that the finitarization of a strict gammoid is obtained by replacing linkages by topological linkages in the definition of independent sets. A topological linkage is a collection of disjoint topological paths, each of which is defined as being the spine of an outgoing comb, a path ending in the centre of a linking fan, or a path ending in B0 . In particular, if a dimaze is {C O , F ∞ }-free, then topological linkages coincide with linkages, leading to the following theorem. ∞
Theorem 3.3 ([12]). Given a dimaze (D, B0 ), independent sets of ML (D, B0 )fin are precisely topologically linkable subsets of V . Consequently, a dimaze defines a finitary strict gammoid if and only if it is {C O , F ∞ }-free.
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Let us turn to cofinitary transversal matroids and strict gammoids. First we recall a construction in [17] that converts a digraph presentation of a finite gammoid to a bipartite graph presentation of its dual transversal matroid. In the following we adjust this construction for our purposes. Given a bipartite graph G = (V , W ) and a matching m0 onto W , we call the pair (G, m0 ) a bimaze, which is an abbreviation for bipartite maze. Definition 3.4. Given a dimaze (D, B0 ), define a bipartite graph D⋆B0 , with bipartition (V , (V \ B0 )⋆ ), where (V \ B0 )⋆ := {v ⋆ : v ∈ V \ B0 } is disjoint from V ; and E (D⋆B0 ) := m0 ∪ {v u⋆ : (u, v) ∈ E (D)}, where m0 := {vv ⋆ : v ∈ V \ B0 }. Call (D, B0 )⋆ := (D⋆B0 , m0 ) the converted bimaze of (D, B0 ). Starting from a dimaze (D, B0 ), we write (V \ B0 )⋆ , m0 and v ⋆ for the corresponding objects in Definition 3.4. The inverse construction is defined as follows: Definition 3.5. Given a bimaze (G, m0 ), where G = (V , W ), define a digraph G⋆m0 such that V (G⋆m0 ) := V and E (G⋆m0 ) := {(v, w) : wv ⋆ ∈ E (G)\ m0 }, where v ⋆ is the vertex in W that is matched by m0 to v ∈ V . Let B0 := V \ V (m0 ). Call (G, m0 )⋆ := (G⋆m0 , B0 ) the converted dimaze of (G, m0 ). Starting from a bimaze (G, m0 ), we write B0 and v ⋆ for the corresponding objects in Definition 3.5 and (V \ B0 )⋆ for W . These constructions are inverse to each other. In particular,
(G, m0 )⋆⋆ = (G, m0 ).
(1)
Given a bimaze (G, m0 ), a set I ⊆ V is m0 -matchable if there is a matching m of I with each connected component of G [m0 ∪ m] finite. The correspondence between linkages in a dimaze and matchings in its converted bimaze is as follows. We remark that the proof of the duality result in [17] relies on the finite counterpart of this correspondence. Lemma 3.6 ([3]). Let (D, B0 ) be a dimaze. Then B is linkable onto B0 in (D, B0 ) if and only if V \ B is m0 -matchable onto (V \ B0 )⋆ in (D, B0 )⋆ . The following is the main tool to prove the two characterization theorems. Theorem 3.7 ([3]). If a dimaze (D, B0 ) is {RI , C A }-free, then ML (D, B0 ) is dual to MT (D⋆B0 ). Similarly, given a bimaze (G, m0 ), if
(G, m0 )⋆ is {RI , C A }-free, then MT (G) is a matroid dual to ML (G, m0 )⋆ .
Recall that any codependent set of a matroid meets every base. Suppose that B is a base and I an independent set such that |B \ I | = |I \ B| < ∞, then one can apply the base exchange axiom [11] to see that I is also a base. Theorem 3.8. Any transversal matroid is cofinitary if and only if it admits a presentation G with a matching m0 of a base such that (G, m0 )⋆ is {RI , C O , C A , F ∞ }-free. Proof. The backward direction follows from Theorems 3.3 and 3.7. Conversely, suppose that M is a cofinitary transversal matroid with a presentation G. Let m0 be a matching of a base V \ B0 of M, then the dimaze (G, m0 )⋆ is RI -free. Suppose not, then there is an incoming ray ending in some b ∈ B0 , which converts to an infinite m0 -alternating walk starting from b in (G, m0 ). This walk gives rise to a matching of (V \ B0 ) + b which contradicts that m0 is a matching of a base of M. For a contradiction, let R be a subdivision of C A in (G, m0 )⋆ . Let I1 be the set of vertices of R with out-degree two, B1 := I1 ∪ (B0 \ V (R)), and v ∈ B0 ∩ V (R). By the construction of R, B1 and B1 + v are both linkable onto B0 . Therefore, by Lemma 3.6 and (1), V \ (B1 + v) is a non-maximal independent set of M. So the complement B1 + v is codependent and contains a cocircuit C of M. Since M is cofinitary, the set C1 := C ∩ I1 is finite. By definition of I1 , C1 is linkable onto some T1 ⊆ B0 \ (B1 + v). Let B2 := C1 ∪ B0 \ T1 . Then |B0 \ B2 | = |B2 \ B0 | < ∞, so that V \ B2 is also a base of M. But this base avoids the cocircuit C . This contradiction shows that (G, m0 )⋆ is C A -free. As (G, m0 )⋆ is {RI , C A }-free, by Theorem 3.7, M ∗ = ML (G, m0 )⋆ . As M ∗ is finitary, (G, m0 )⋆ does not contain any subdivision of C O or F ∞ either. Together with Theorem 3.7, we have the following. Corollary 3.9. Any cofinitary transversal matroid is dual to a strict gammoid. Next, we turn to cofinitary strict gammoids. Theorem 3.10. Any strict gammoid is cofinitary if and only if it admits a {C A , RI }-free presentation such that the in-degree of each vertex is finite.
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Proof. Let M be a strict gammoid. Suppose it has a {C A , RI }-free presentation (D, B0 ) where the in-degree of every vertex is finite. By Theorem 3.7, M ∗ = MT (D⋆B0 ). As the in-degree of D is finite, D⋆B0 is left locally finite. So by Proposition 3.2, MT (D⋆B0 ) is finitary and hence M is cofinitary. Conversely, suppose that M is a cofinitary and admits (D, B0 ) as a presentation. As all the vertices in D linkable to a fixed vertex in B0 form a cocircuit, (D, B0 ) is RI -free. The same reason implies also that every vertex with infinite in-degree cannot be linked to B0 , hence, such a vertex is a loop. Therefore, deleting all the edges incident in such a vertex results in another presentation (D′ , B0 ) of M. For a contradiction, let R be a subdivision of C A in (D′ , B0 ). Let I1 be the set of vertices of R with out-degree two and B1 := I1 ∪ (B0 \ V (R)). Since B1 is a non-maximal independent set, V \ B1 is codependent and contains a cocircuit C of M. Since M is cofinitary, C ∩ B0 is a finite set. Then B0 \ C can be extended by some |C ∩ B0 | vertices of I1 to a base disjoint from C which contradicts the codependence of C . This contradiction shows that (D′ , B0 ) is C A -free. Analogous to Corollary 3.9, applying Theorem 3.7, we have the following. Corollary 3.11. Any cofinitary strict gammoid is dual to a transversal matroid. We remark that neither Corollary 3.9 nor Corollary 3.11 remains true when we drop the cofinitary assumption (see [3]). 4. Nearly finitary gammoids Classical matroid theory usually studies matroids without infinite circuits. On the other hand, recent investigations have led to a growing list of matroids that do contain infinite circuits. Finitarizing a matroid is a process to disregard infinite circuits, thereby relates new infinite matroids to classical ones. Thus, for example (see [2]), the finitarization of a thin sums matroid is a representable matroid, whereas that of a bond matroid of a graph G is the finite bond matroid of G [10]. Nearly finitary matroid is a bridge between finitary and general infinite matroids in the sense that it is ‘only finitely far away’ from a finitary matroid. A matroid M is called nearly finitary [6] if we can delete finitely many points from every base of the finitarization M fin to get a base of M. Equivalently, every base can be extended to a base of the finitarization by adding finitely many points. If the number of deletions in the former, or equivalently, additions in the latter is bounded by an integer k ≥ 0, then M is called k-nearly finitary. Note that in a nearly finitary matroid the number of deletions or additions may depend on the base. For example, in the algebraic cycle matroid of the one-way infinite ladder (for more information, see [11]), a spanning tree without double ray is a base of the matroid as well as its finitarization, whereas two disjoint rays spanning the graph constitute a base, to which an edge can be added without forming any finite cycle. The problem [7] to determine whether a nearly finitary matroid is k-nearly finitary for some k ≥ 0 is open. We will solve this problem for strict gammoids and transversal matroids. For strict gammoids, we are going to use Theorem 3.3 and the following modified version of a result of Halin [15]. We mimic a proof of Andreae [13, Theorem 8.2.5]. Given a path P = x0 · · · xi · · · xj · · · xn , let xi P := xi · · · xn , Pxi := x0 · · · xi , xi Pxj := xi · · · xj and x˚i P := xi+1 · · · xn . For a ray R with an initial vertex, xi R, Rxi , xi Rxj and x˚i R are defined analogously. Given an outgoing comb C and v on its spine R, we write C v for the graph obtained by taking the union of the initial segment Rv of R and the spikes of C that meet Rv . C v is called an initial segment of C . Given two outgoing combs C and C ′ and the initial segments C v and C ′ v ′ , we say that C v is a proper initial segment of C ′ v ′ if Rv is a proper initial segment of R′ v ′ and the set of spikes of C v is a proper subset of that of C ′ v ′ . Lemma 4.1. If for every k ∈ N there is a set of spine-disjoint outgoing combs of size k, then there is an infinite such set. Proof. For any outgoing comb Cin , let Rni be its spine. We construct our infinite set of outgoing combs inductively. In step n ≥ 0, we shall find n spine-disjoint outgoing combs C1n , . . . , Cnn and choose initial segments Cin xni of them. In step n + 1, we +1 n +1 +1 n +1 n +1 n n choose the outgoing combs C1n+1 , . . . , Cnn+ , . . . , xnn+ xi . Then the 1 and x1 1 so that Ci xi is a proper initial segment of Ci graph Ci∗ := n∈N Cin xni is an outgoing comb and (Ci∗ )i∈N is an infinite family of spine-disjoint outgoing combs. We start from the empty set of initial segments of outgoing combs. Suppose that, in step n, C1n , . . . , Cnn and their initial segments C1n xn1 , . . . , Cnn xnn are given, then the construction at step n + 1 is as follows. Abbreviate Cin , Rni and xni to Ci , Ri and xi , respectively. Let C be any set of n2 + 1 spine-disjoint outgoing combs that do not meet any of C1 x1 , . . . , Cn xn . For 1 ≤ i ≤ n, if x˚i Ri meets at most n of the spines of outgoing combs in C , we delete those outgoing combs from C , put Cin+1 := Ci , and choose as xni +1 the first vertex on x˚i Ri where Ci has a spike. Let I := {i ≤ n : Cin+1 is still undefined}, and put m := |I |. Then C still contains at least n2 + 1 − (n − m)n ≥ m2 + 1 outgoing combs. For every i ∈ I, let zi be a vertex on Ri such that xi Ri zi meets exactly m distinct spines of outgoing combs in C . Then Z := i∈I xi Ri zi meets at most m2 spines of +1 the outgoing combs in C . Delete the other outgoing combs from C and choose one of these deleted combs as Cnn+ 1 together 1 with an arbitrary xnn+ +1 on its spine. For each outgoing comb in C , we choose on the spine R a vertex y(R) beyond the vertices of R in Z such that there is a spike at y(R). Let Y := {y(R) : R is the spine of an outgoing comb in C }. Then X := {xi : i ∈ I } clearly cannot be separated from Y in the union H of Z and the paths Ry by fewer than m vertices. Apply Menger’s theorem to find m disjoint X –Y paths
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Pi = xi · · · yi (i ∈ I ) in H. For each i ∈ I, let Ci′ denote the outgoing comb from C whose spine contains yi . Now choose as Cin+1 the outgoing comb obtained by concatenating Ci xi , Pi and yi Ci′ , and put xni +1 := yi . Proposition 4.2. A nearly finitary strict gammoid is k-nearly finitary for some k ≥ 0. Proof. Let M = ML (D, B0 ) be a nearly finitary strict gammoid. By Theorem 3.3, there cannot be infinitely many spinedisjoint outgoing combs, since otherwise the union of the initial vertices of their spines with B0 is independent in M fin , thereby rendering M not nearly finitary. It follows then, by Lemma 4.1, that the size of any set of spine-disjoint outgoing combs is bounded. Similarly, there cannot be infinitely many vertices which are the centre of a linking fan, otherwise, B0 together with these centres form an independent set in M fin . Let k be the sum of the maximum number of spine-disjoint outgoing combs and the number of centres of linking fans. Let B be an independent set in M fin . By Theorem 3.3, there is a topological linkage of B to B0 . In such a linkage, vertices may be the initial vertex of the spine of an outgoing comb or the initial vertex of a path leading to the centre of a linking fan, or a path to B0 . By our estimate B contains at most k vertices of the first two types. Deleting these vertices leaves us with a linkable subset of B. Thus, M is k-nearly finitary. Next, we will prove that any nearly finitary transversal matroid is k-nearly finitary for some k. In fact, our proof depends only on the property that no infinite circuit is contained in a union of finite circuits. Note that a transversal matroid M satisfies this property. Indeed, suppose M contains an infinite circuit C and M = MT (G) for some bipartite graph G. Then G − (V \ C ) is a presentation of the circuit C . By the contrapositive of Proposition 3.2, there is a vertex in C having infinite degree in G. Since such a vertex cannot lie in any finite circuit of M, C is not contained in a union of finite circuits. Lemma 4.3. Let M be a matroid such that no infinite circuit is contained in a union of finite circuits. If M is nearly finitary, then M is k-nearly finitary for some integer k ≥ 0. Proof. Let L be the set of coloops of M fin . Then each element in M \ L is contained in a finite circuit. By the assumption on M, M \ L does not contain any infinite circuit. Given a base B f of M fin , then L ⊆ B f and B f \ L is a base of the finitary matroid M \ L, since B f \ L does not contain any finite circuit of M and addition of any element in M \ B f creates one. Let B1 be a base of M .L. Then B = B1 ∪ (B f \ L) is a base of M. Since M is nearly finitary, B f \ B is finite. On the other hand, B f \ B = L \ B1 . As B f was arbitrary, M is k-nearly finitary for k := |L \ B1 |. Remark 4.4. In fact, if B is a base of M and B f is a base of M fin containing B, then |B f \ B| = k. Indeed, B ∩ L contains a base B1 of M .L, and B \ L can be extended in B f \ L to a base B2 of M \ L. In fact, B2 = B f \ L, so |B f \ B| = |B f \ (B1 ∪ B2 )| = k. In case M is a transversal matroid, the extension of a base of M to one of M fin occurs in the set of vertices of infinite degree. Hence, we have the following. Proposition 4.5. For any integer k ≥ 0 and any bipartite graph G = (V , W ) such that there are at most k vertices of infinite degree in V , MT (G) is a k-nearly finitary matroid. The converse is true for k ≤ 1, that is, a k-nearly finitary transversal matroid admits a presentation with at most k vertices in V of infinite degree. For k = 0, we apply Proposition 3.2. For k = 1, let L be the set of vertices of infinite degree of V in G. We may assume that a base BL of MT (G).L misses exactly one vertex v in L. We form a graph G′ by moving all edges on BL to v , except a matching m of a base of MT (G) extending BL , i.e. we replace each edge e = uw ∈ E (G) − m with u ∈ BL − v by the edge vw . A long case analysis then shows that G′ defines the same transversal matroid as MT (G). We omit the proof. On the other hand, a 2-nearly finitary transversal matroid may only have presentations with infinitely many vertices of infinite degree. Example 4.6. The circuits of the transversal matroid M defined by the bipartite graph in Fig. 1 are {V − x1 , V − x2 , V \ Vi : i ∈ N}, where Vi = {vij : j ∈ N}. For any presentation G = (V , W ) of M, there are infinitely many vertices of infinite degree in V . Proof. It is straightforward to compute the set of circuits. Let X = {x1 , x2 }. Fix a matching m of the base V \ X . Let (D, X ) = (G, m)⋆ . It suffices to prove that in D, there is a vertex of infinite in-degree in each Vi . Note that an (infinite) m-alternating walk in G corresponds to an (infinite) directed path in D. As V − x1 and V − x2 are circuits, there is no incoming ray to X in D. Since V \ Vi is a circuit, there are no two disjoint m-alternating walks from X to Vi in G. Therefore, there are no two disjoint paths from Vi to X in D. By Menger’s theorem, Vi is separated from X by a vertex vi . For any v ∈ Vi , v ′ ∈ Vj where i ̸= j, as V \ {v, v ′ } is matchable, there is a pair of disjoint paths from {v, v ′ } to X . It follows that vi ∈ Vi . For any v ∈ Vi − vi , if there is an edge (v, v ′ ) with v ′ ∈ Vj , then the edge can be extended to a path from v to X avoiding vi , using a path in a linkage from {v ′ , vi } to X , which is a contradiction. Hence, within Vi , there is a path from v to vi . As there is no incoming ray to X , it follows that Vi contains a vertex of infinite in-degree. So having at most k vertices of infinite degree is sufficient but not necessary for a transversal matroid to be k-nearly finitary. However, we can delete some k vertices of a k-nearly finitary transversal matroid to get a finitary minor. To be precise, we have the following.
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Fig. 1. A 2-nearly finitary transversal matroid.
Proposition 4.7. Let M be a matroid such that no infinite circuit is contained in a union of finite circuits, then M is k-nearly finitary if and only if there is a set X of k points such that M \ X is finitary. Proof. Suppose that M is k-nearly finitary. Let L be the set of coloops of M fin . By the assumption on M, M \ L is a finitary matroid. Let B1 be a base of M .L and B2 be a base of M \ L. We claim that X := L \ B1 is the required set. As B1 ∪ B2 ∪ X is a base of M fin , |X | ≤ k. It remains to show that M \ X is finitary. Let C be a circuit of M \ X . As C ⊆ B1 ∪ (C \ L) and B1 is a base of M .L, C ∩ L = ∅. So C is a circuit of the finitary matroid M \ L. Conversely, let X be any set of size at most k such that M \ X is finitary and let Bfin be a base of M fin . As any infinite circuit must meet X , Bfin \ X is independent in M, and we can extend it inside X ∩ Bfin to a base B′ of M. As Bfin is arbitrary and |Bfin \ B′ | ≤ |X | ≤ k, M is k-nearly finitary. Note that for the backward direction we do not need any assumption on M. Finally, Lemma 4.3 and the paragraph before that together with Proposition 4.7 imply the following. Proposition 4.8. Let M be a transversal matroid. The following are equivalent:
• M is nearly finitary. • M is k-nearly finitary, for some integer k ≥ 0. • There exists a set X of k vertices such that M \ X is finitary. A consequence of Theorem 3.3 is that a strict gammoid is k-nearly finitary if and only if each of its presentations contains a total of at most k spine-disjoint outgoing combs and centres of linking fans. In line with our interest in graph theoretic characterization, it would be interesting to answer the following. Problem 4.9. Characterize k-nearly finitary transversal matroids in graph theoretic terms. 5. Representability of gammoids It was proved in [25] that any finite transversal matroid (and hence any finite strict gammoid by duality) is representable over any large enough field. We first use the arguments in [18,19] to sketch a proof that any finitary gammoid is representable. For a function f : E → kA for some field k and set A, let M (f ) denote the vector matroid generated by f , i.e. I(M (f )) = {I ⊆ E : f (I ) is a linearly independent set of vectors over k}. Proposition 5.1. A finitary transversal matroid is representable, and so is a finitary strict gammoid. Proof. We need only to prove that a finitary strict gammoid ML (D, B0 ) is representable, since a finitary transversal matroid is the restriction of some finitary strict gammoid. Let Y = {yij : (i, j) ∈ E (D)} be a set of algebraically independent indeterminates. Let kY be the field of fractions of the formal power series of Y over the rationals. Associate to each edge B (i, j) the weight yij . Define g : V → kY0 such that gv (b) is the sum of the weight of all the directed walks from v to b ∈ B0 , where the weight of any such walk is the product of the weight of its edges.
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Fig. 2. A wild transversal matroid (where the bold edges form a matching of a base) and its strict gammoid dual.
If I is a finite independent set, then there is a linkage from I onto some B ⊆ B0 . By expanding the determinant of the square matrix defined by g with columns in I and rows in B as in [18], we conclude that I is independent in M (g ). Since both ML (D, B0 ) and M (g ) are finitary, this shows that any independent set in ML (D, B0 ) is also independent in M (g ). On the other hand, let C be a circuit in ML (D, B0 ). By Menger’s theorem, there is a separator S of size |C | − 1 separating ′ ′′ ′ C from B0 . Note that for v ∈ C , gv (b) = s∈S gv (s)gs (b) where gv (s) is the sum of weights of walks from v to s ∈ S in D and gs′′ (b) is the sum of weights of walks from s to b in the modified digraph obtained from D by deleting the edges entering S. Since |S | < |C |, there is a nonzero vector (λv : v ∈ C ) such that v∈C λv gv′ (s) = 0 for any s ∈ S. Hence, v∈C λv gv (b) = 0 for all b ∈ B0 . Thus, C is dependent in M (g ). We conclude that ML (D, B0 ) = M (g ). By definition, a representable matroid is finitary. In order to capture non-finitary matroids, a generalization called thin sums representability was proposed in [10]. Given sets E , A and a field k and a function f : E → kA mapping e to fe : A → k, a thin dependence is a function c : E → k such that for each a ∈ A, {e ∈ E : c (e)fe (a) ̸= 0} is finite and e∈E c (e)fe (a) = 0. Let Mts (f ) denote the pair E with the collection of subsets of E each of which does not admit a nonzero thin dependence. In case Mts (f ) is a matroid, it is called a thin sums matroid, and any matroid isomorphic to Mts (f ) for some f : E → kA thin sums representable. It was proved in [2] that a representable matroid is always thin sums representable. While it is not true that Mts (f ) is a matroid for every f , it is the case when f is thin, i.e. {e ∈ E : fe (a) ̸= 0} is finite for each a. In fact, Mts (f ) is cofinitary if f is thin [2]. A matroid is tame if the intersection of a circuit and a cocircuit is always finite, wild otherwise [8]. Since the class of tame thin sums matroids is closed under duality [2], we have the following corollary by applying Corollaries 3.9 and 3.11. Corollary 5.2. A cofinitary transversal matroid is thin sums representable, and so is a cofinitary strict gammoid. It is not true that every transversal matroid or strict gammoid is thin sums representable over some field. We will show that the bipartite graph in Fig. 2 defines a wild transversal matroid that is not thin sums representable over any field. We note that the bipartite graph does not contain any ray. For a proof that any bipartite graph G not containing any ray defines a transversal matroid, see [1, Lemma 3.6.19]. Essentially the proof of (IM) makes use of an infinite version of König’s duality theorem [5] and the fact that the vertices of a matching of G given by this theorem which are in the ground set form a maximal independent set of MT (G) as G does not contain any ray. Lemma 5.3. Let M be a thin sums matroid whose ground set is covered by finitely many circuits. Then M is cofinitary (and hence is not wild). Proof. Let M = Mts (f ) where f : E → kA for some field k, set A and function f . It suffices to show that f is thin, for then we conclude by [2] that Mts (f ) is cofinitary. Observe that for any circuit C , there is a thin dependence c whose support is precisely C . By definition of thin dependence, for each a, {e ∈ E : c (e)fe (a) ̸= 0} = {e ∈ E : fe (a) ̸= 0} ∩ C is finite. It follows that {e ∈ E : fe (a) ̸= 0} is finite, since E is a union of finitely many circuits. Therefore, f is thin as desired. Proposition 5.4. Let G be a bipartite graph such that V (G) = {x, y, ai , bi : i ≥ 1} ∪ {Z , Ai : i ≥ 1} and E (G) = {ai Ai , ai Z , bi Ai , xAi , yAi , xZ , yZ : i ≥ 1} (see Fig. 2). Then MT (G) is a wild transversal matroid not thin sums representable over any field. Moreover, MT∗ (G) is a wild strict gammoid not thin sums representable over any field. Proof. It can be checked that the infinite set {ai : i ≥ 1} + x + y is both a circuit and a cocircuit (and so is {bi : i ≥ 1} + x + y). Hence, MT (G) is wild. Moreover, as these two circuits cover the ground set, by Lemma 5.3, MT (G) is not thin sums representable. By Theorem 3.7, MT∗ (G) is a strict gammoid. Again by Lemma 5.3, MT∗ (G) is not thin sums representable either.
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In view of Propositions 5.1 and 5.2, we propose the following. Conjecture 5.5. A tame gammoid is thin sums representable. Acknowledgements The authors thank the anonymous referees for their valuable comments in a submission of this paper. HFL thanks the support from the Alexander von Humboldt Foundation and the Croucher Foundation. References [1] S.H. Afzali Borujeni, Representability of infinite matroids and the structure of linkages in digraphs (Ph.D. thesis), Universität Hamburg, Germany, 2014, http://ediss.sub.uni-hamburg.de/volltexte/2014/7030/. [2] S.H. Afzali Borujeni, N. Bowler, Thin sums matroids and duality, Adv. Math. 271 (2015) 1–29. [3] S.H. Afzali Borujeni, H.-F. Law, M. Müller, Infinite gammoids: Minors and duality (2014). arXiv:1411.2277v1. [4] S.H. Afzali Borujeni, H.-F. Law, M. Müller, Infinite gammoids, Electron. J. Combin. 22 (2015) # P1.53. [5] R. Aharoni, König’s duality theorem for infinite bipartite graphs, J. Lond. Math. Soc. 29 (1984) 1–12. [6] E. Aigner-Horev, J. Carmesin, J.-O. Fröhlich, Infinite matroid union (2011). arXiv:1111.0602v2. [7] E. Aigner-Horev, J. Carmesin, J.-O. Fröhlich, On the intersection of infinite matroids (2011). arXiv:1111.0606. [8] N. Bowler, J. Carmesin, Matroids with an infinite circuit-cocircuit intersection, J. Combin. Theory Ser. B 107 (2014) 78–91. [9] R.A. Brualdi, E.B. Scrimger, Exchange systems, matchings and transversals, J. Combin. Theory 5 (1968) 244–257. [10] H. Bruhn, R. Diestel, Infinite matroids in graphs, in: Infinite Graph Theory Special Volume of Discrete Math., vol. 311, 2011, pp. 1461–1471. [11] H. Bruhn, R. Diestel, M. Kriesell, R. Pendavingh, P. Wollan, Axioms for infinite matroids, Adv. Math 239 (2013) 18–46. [12] J. Carmesin, Topological infinite gammoids, and a new Menger-type theorem for infinite graphs (2014). arXiv:1404.0151v1. [13] R. Diestel, Graph Theory, fourth ed., Springer, Heidelberg, 2010. [14] J. Edmonds, D.R. Fulkerson, Transversals and matroid partition, J. Res. Natl. Bur. Stand. 69B (1965) 147–153. [15] R. Halin, Über die Maximalzahl fremder unendlicher Wege (in German), Matt. Nachr. 30 (1965) 63–85. [16] D.A. Higgs, Matroids and duality, Colloq. Math. 20 (1969) 215–220. [17] A.W. Ingleton, M.J. Piff, Gammoids and transversal matroids, J. Combin. Theory Ser. B 15 (1973) 51–68. [18] B. Lindström, On the vector representations of induced matroids, Bull. Lond. Math. Soc. 5 (1973) 85–90. [19] J.H. Mason, On a class of matroids arising from paths in graphs, Proc. Lond. Math. Soc. 25 (1972) 55–74. [20] L. Mirsky, H. Perfect, Applications of the notion of independence to problems of combinatorial analysis, J. Combin. Theory 2 (1967) 327–357. [21] J.G. Oxley, Infinite matroids, Proc. Lond. Math. Soc. 37 (1978) 259–272. [22] J.G. Oxley, Some problems in combinatorial geometries (Ph.D. thesis), Oxford University, U.K, 1978. [23] J.G. Oxley, Matroid Theory, Oxford University Press, 1992. [24] H. Perfect, Applications of Menger’s graph theorem, J. Math. Anal. Appl. 22 (1968) 96–111. [25] M.J. Piff, D.J.A. Welsh, On the vector representation of matroids, J. Lond. Math. Soc. 2 (1970) 284–288.