Infinite gammoids

Seyed Hadi, Afzali Borujeni, Hiu-Fai Law, Malte Müller
2015 the electronic journal of combinatorics   unpublished
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
more » ... rove 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. 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 also Higgs [13] and Oxley [15]). Our contribution to the development focusses on the class of gammoids. In this first paper, the main object of investigation is the bases of infinite strict gammoids. (The second one considers other aspects including duality and minors [2].) The concept of gammoids originated from the transversal matroids of Edmonds and Fulkerson [11]. 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. Perfect [17] introduced the class of gammoids by replacing matchings in bipartite graphs with disjoint directed paths in digraphs. Later, Mason [14] started the study of a subclass of gammoids known as strict gammoids.