Disjoint dijoins for classes of dicuts in finite and infinite digraphs

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 B of dicuts of a directed graph the maximum number of disjoint sets of edges meeting every dicut in B equals the size of a minimum dicut in 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. §1. IntroductionIn this paper we consider directed graphs, which we briefly denote as digraphs. A dicut in a digraph 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 [16] 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 of edges are called dijoins.Woodall conjectured the following in some sense dual statement, where the roles of the minimum and maximum are reversed.Conjecture 1 (Woodall 1976 [25]). The size of a smallest non-empty dicut in a finite digraph D is equal to the size of the largest set of disjoint dijoins of D.