Abstract
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.
Highlights
Infinite matroid theory has recently seen a surge of activity (e.g. [1], [8], [11]), after Bruhn et al [10] found axiomatizations of infinite matroids with duality, solving a longstanding problem of Rado [23]
The electronic journal of combinatorics 22(4) (2015), #P4.37 that finite strict gammoids and finite transversal matroids are dual to each other, a key fact to their proof that the class of finite gammoids is closed under duality
If we want to show that a minor of a gammoid is a gammoid, it suffices to show that a contraction minor M/S of a strict gammoid M is a strict gammoid; where S may be assumed to be independent by Lemma 2.2
Summary
Infinite matroid theory has recently seen a surge of activity (e.g. [1], [8], [11]), after Bruhn et al [10] found axiomatizations of infinite matroids with duality, solving a longstanding problem of Rado [23]. A standard proof of the fact that finite gammoids are minor-closed as a class of matroids proceeds via duality The proof of this fact can be extended to infinite dimazes whose underlying (undirected) graph does not contain any ray, but it breaks down when rays are allowed. The electronic journal of combinatorics 22(4) (2015), #P4.37 that finite strict gammoids and finite transversal matroids are dual to each other, a key fact to their proof that the class of finite gammoids is closed under duality. As we will see in Example 4.12, there is a strict gammoid whose dual is not a path-transversal matroid This strict gammoid has the property that any dimaze defining it contains an alternating comb, a dimaze that is studied to some details in [2]. We remark that the theorem is used in [3] to characterize cofinitary transversal matroids and cofinitary strict gammoids
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