Disjoint dijoins for classes of dicuts in finite and infinite digraphs

TL;DR

This paper investigates a generalized version of a classical graph theory problem, focusing on disjoint dijoins and dicuts in both finite and infinite directed graphs, and explores related conjectures and special classes of dicuts.

Contribution

It extends the theory of dijoins and dicuts to specific classes, including infinite dicuts, and examines the duality conjecture in these contexts.

Findings

Verified the conjecture for nested classes of finite dicuts

Confirmed the conjecture for dicuts of minimum size

Explored the relationship with capacitated versions of the problem

Abstract

A dicut in a directed graph is a cut for which all of its edges are directed to a common side of the cut. A famous theorem of Lucchesi and Younger states that in every finite digraph the least size of a set of edges meeting every non-empty dicut equals the maximum number of disjoint dicuts in that digraph. Such sets are called dijoins. Woodall conjectured a dual statement. He asked whether the maximum number of disjoint dijoins in a directed graph equals the minimum size of a non-empty dicut. We study a modification of this question where we restrict our attention to certain classes of non-empty dicuts, i.e. whether for a class $\mathfrak{B}$ of dicuts of a directed graph the maximum number of disjoint sets of edges meeting every dicut in $\mathfrak{B}$ equals the size of a minimum dicut in $\mathfrak{B}$. In particular, we verify this questions for nested classes of finite dicuts, for the class of dicuts of minimum size, and for classes of infinite dibonds, and we investigate how this generalised setting relates to a capacitated version of this question.