Duality Respecting Representations and Compatible Complexity Measures for Gammoids

TL;DR

This paper introduces special digraph representations for gammoids that facilitate duality operations and define a complexity measure, ensuring classes with bounded complexity are closed under duality, minors, and sums.

Contribution

It presents a novel class of digraph representations for gammoids that simplify duality and introduces a compatible complexity measure with closure properties.

Findings

Dual representations allow easy dual gammoid construction

A complexity measure for gammoids is defined and shown to be closed under key operations

Special representations enable duality and minor operations to commute

Abstract

We show that every gammoid has special digraph representations, such that a representation of the dual of the gammoid may be easily obtained by reversing all arcs. In an informal sense, the duality notion of a poset applied to the digraph of a special representation of a gammoid commutes with the operation of forming the dual of that gammoid. We use these special representations in order to define a complexity measure for gammoids, such that the classes of gammoids with bounded complexity are closed under duality, minors, and direct sums.