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# Pascal Gollin, Disjoint dijoins for classes of dibonds in finite and infinite digraphs

## Tuesday, April 7, 2020 @ 4:30 PM - 5:30 PM KST

A *dibond* in a directed graph is a bond (i.e. a minimal non-empty 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 dibonds in that digraph. We call such sets *dijoins*.

Woodall conjectured a dual statement. He asked whether the maximum number of disjoint dijoins in a digraph equals the minimum size of a dibond.

We study a modification of this question where we restrict our attention to certain classes of dibonds, i.e. whether for a class $\mathfrak{B}$ of dibonds of a digraph the maximum number of disjoint edge sets meeting every dibond in $\mathfrak{B}$ equal the size a minimum dibond in $\mathfrak{B}$.

In particular, we verify this questions for nested classes of dibonds, for the class of dibonds of minimum size, and for classes of infinite dibonds.