Abstract
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.
Highlights
Infinite matroid theory has seen vigorous development (e.g. [1], [5] and [7]) since Bruhn et al [6] in 2010 gave five equivalent sets of axioms for infinite matroids in response to a problem proposed by Rado [19]
A transversal matroid can be defined by taking as its independent sets the subsets of a fixed vertex class of a bipartite graph matchable to the other vertex class
The vertex sets linkable onto the exits form the bases of a matroid if and only if the dimaze contains no alternating comb
Summary
Infinite matroid theory has seen vigorous development (e.g. [1], [5] and [7]) since Bruhn et al [6] in 2010 gave five equivalent sets of axioms for infinite matroids in response to a problem proposed by Rado [19] (see Higgs [13] and Oxley [15]). A set of vertices of (the digraph of) the dimaze is independent if it is linkable to the exits by a collection of disjoint directed paths. While a dimaze containing an alternating comb may still define a matroid, the set of bases is a proper subset of the sets linkable onto the exits and can be difficult to describe. We answer this negatively via the intermediate step of showing that any tree, when viewed as a bipartite graph, defines a transversal matroid, a statement which is of independent interest
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