Hitting cycles through prescribed vertices or edges

TL;DR

This paper establishes a linear bound between directed tree-width and cycle-width, providing new insights into cycle hitting sets in directed and bidirected graphs, with some open problems remaining.

Contribution

It proves a linear bound between directed tree-width and cycle-width and extends cycle hitting set results to bidirected graphs, improving previous bounds.

Findings

Directed tree-width is linearly bounded by cycle-width.

Cycle hitting set bounds hold in bidirected graphs.

Edge-variant cycle hitting results hold in undirected and directed graphs.

Abstract

We prove that for every set $S$ of vertices of a directed graph $D$, the maximum number of vertices in $S$ contained in a collection of vertex-disjoint cycles in $D$ is at least the minimum size of a set of vertices that hits all cycles containing a vertex of $S$. As a consequence, the directed tree-width of a directed graph is linearly bounded in its cycle-width, which improves the previously known quadratic upper bound. We further show that the corresponding statement in bidirected graphs is true and that its edge-variant holds in both undirected and directed graphs, but fails in bidirected graphs. The vertex-version in undirected graphs remains an open problem.