Abstract
AbstractA 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 an edge set meeting every dicut equals the maximum number of disjoint dicuts in that digraph. In this first paper out of a series of two papers, we conjecture a version of this theorem using a more structural description of this min‐max property for finite dicuts in infinite digraphs. We show that this conjecture can be reduced to countable digraphs where the underlying undirected graph is 2‐connected, and we prove several special cases of the conjecture.
Highlights
In finite structural graph theory there are a lot of theorems which illustrate the dual nature of certain objects by relating the maximum number of disjoint objects of a certain type in a graph with the minimal size of an object of another type in that graph
A famous theorem of Lucchesi and Younger states that in every finite digraph the least size of an edge set meeting every dicut equals the maximum number of disjoint dicuts in that digraph
In this first paper out of a series of two papers, we conjecture a version of this theorem using a more structural description of this min‐max property for finite dicuts in infinite digraphs
Summary
In finite structural graph theory there are a lot of theorems which illustrate the dual nature of certain objects by relating the maximum number of disjoint objects of a certain type in a graph with the minimal size of an object of another type in that graph. In a weakly connected digraph D we call an edge set F ⊆ E (D) a finitary dijoin of D if it intersects every nonempty finite dicut of D. Building up on this definition, we call a tuple (F, ) as in Theorem 1.2 but where F is a finitary dijoin and is a set of disjoint finite dicuts of D, an optimal pair for D.
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