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Hitting cycles through prescribed vertices or edges

Nathan Bowler, Ebrahim Ghorbani, Florian Gut, Raphael W. Jacobs, Florian Reich

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.

Hitting cycles through prescribed vertices or edges

Abstract

We prove that for every set of vertices of a directed graph , the maximum number of vertices in contained in a collection of vertex-disjoint cycles in is at least the minimum size of a set of vertices that hits all cycles containing a vertex of . 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.

Paper Structure

This paper contains 8 sections, 16 theorems, 18 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Proofs of the main results
  3. The edge-version in undirected graphs
  4. The edge-version in directed graphs
  5. The vertex-version in bidirected graphs
  6. The vertex-version in directed graphs
  7. A counterexample to the edge-version in bidirected graphs
  8. An Application: (Bi)directed Tree-Width and Cycle-Width

Key Result

Theorem 1.1

There exists $c \in \mathbb N$ such that, for each $k \in \mathbb N$, every undirected graph $G$ contains either $k$ vertex-disjoint cycles or a set $X$ of at most $c\, k \log k$ vertices that hits every cycle of $G$.

Figures (2)

  • Figure 1: A bidirected graph for which the columns in its incidence matrix sum up to zero because each vertex has an equal number of positive and negative incident edges, yet it has no bidirected cycle.
  • Figure 2: A counterexample to the edge-version in bidirected graphs. More precisely, a bidirected graph in which there are no two edge-disjoint cycles containing a dotted edge. For every edge set $X$ of size $\leq 2$ there exists a cycle in $B - X$ containing a dotted edge.

Theorems & Definitions (24)

  • Theorem 1.1: erdos1965independentcircuitscontained
  • Theorem 1.2: reed1996packingdirectedcircuits
  • Theorem 1.3: pontecorvi2012disjointcyclesintersecting
  • Corollary 1.4
  • Theorem 2.1
  • proof
  • Theorem 2.2
  • proof
  • Theorem 2.3
  • Remark 2.4
  • ...and 14 more