Infinite Gammoids: Minors and Duality

TL;DR

This paper explores properties of infinite gammoids, including minors and duality, establishing conditions under which they are minor-closed and characterizing their duals, extending finite gammoid theory to the infinite case.

Contribution

It proves that certain classes of infinite gammoids are minor-closed and characterizes their duals, extending finite gammoid duality concepts to the infinite setting.

Findings

Certain infinite gammoid classes are minor-closed.

Finite-rank minors of gammoids are gammoids.

Topological gammoids coincide with finitary gammoids.

Abstract

This sequel to our paper (Infinite gammoids, 2014) considers minors and duals of infinite gammoids. We prove that a class of gammoids definable 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 introduced by Carmesin (Topological infinite gammoids, and a new Menger-type theorem for infinite graphs, 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 alternating-comb-free strict gammoids, introduced in the prequel, 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.