Hitting cycles through prescribed vertices or edges

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.

💡 Research Summary

The paper investigates a quantitative duality between feedback sets (vertex or edge hitting sets) and collections of vertex‑disjoint cycles that are required to intersect a prescribed set S of vertices (or a prescribed set F of edges). Classical results such as the Erdős‑Pósa theorem (for undirected graphs) and Younger’s conjecture (proved by Reed, Robertson, Seymour, and Thomas for directed graphs) give a logarithmic‑linear trade‑off between the number k of disjoint cycles and the size of a feedback vertex set that hits all cycles. However, when the cycles are required to contain vertices from a specific set S, this trade‑off no longer holds in general: there are graphs where at most k S‑cycles exist while any feedback vertex set for S‑cycles must have size Ω(k log k).

Motivated by this, the authors pose Question 1.4: for a given graph (undirected, directed, or bidirected) and a prescribed set S (or F for edges), does the maximum number of prescribed vertices (or edges) that can be covered by a collection of vertex‑disjoint (or edge‑disjoint) cycles dominate the minimum size of a hitting set for all S‑cycles (or F‑cycles)? In other words, is the inequality

 max |S ∩ V(C)| ≥ min |X|

true, where the left side ranges over collections C of vertex‑disjoint cycles and the right side over subsets X that intersect every S‑cycle?

The paper answers this question for several variants:

1. Edge‑version in undirected graphs (Theorem 2.1).
   Using the cycle space over ℤ₂, the authors construct the incidence matrix M between edges and cycles. They show that the column space of M contains a vector whose restriction to the prescribed edge set F has all entries 1, provided the rank of the submatrix M_F is at least the size t of a minimum hitting set. This yields the desired inequality.

2. Edge‑version in directed graphs (Theorem 2.2).
   The proof relies on linear programming duality. The primal LP maximises the number of prescribed edges covered by a collection of edge‑disjoint directed cycles; the constraint matrix A =