Packing directed circuits quarter-integrally

07/04/2019
by   Tomáš Masařík, et al.
0

The celebrated Erdős-Pósa theorem states that every undirected graph that does not admit a family of k vertex-disjoint cycles contains a feedback vertex set (a set of vertices hitting all cycles in the graph) of size O(k k). After being known for long as Younger's conjecture, a similar statement for directed graphs has been proven in 1996 by Reed, Robertson, Seymour, and Thomas. However, in their proof, the dependency of the size of the feedback vertex set on the size of vertex-disjoint cycle packing is not elementary. We show that if we compare the size of a minimum feedback vertex set in a directed graph with quarter-integral cycle packing number, we obtain a polynomial bound. More precisely, we show that if in a directed graph G there is no family of k cycles such that every vertex of G is in at most four of the cycles, then there exists a feedback vertex set in G of size O(k^4). On the way there we prove a more general result about quarter-integral packing of subgraphs of high directed treewidth: for every pair of positive integers a and b, if a directed graph G has directed treewidth Ω(a^6 b^8^2(ab)), then one can find in G a family of a subgraphs, each of directed treewidth at least b, such that every vertex of G is in at most four subgraphs.

READ FULL TEXT

page 1

page 2

page 3

page 4

research
02/20/2018

The Parameterized Complexity of Packing Arc-Disjoint Cycles in Tournaments

Given a directed graph D on n vertices and a positive integer k, the Arc...
research
08/05/2020

Constant Congestion Brambles

A bramble in an undirected graph G is a family of connected subgraphs of...
research
03/07/2019

Colouring Non-Even Digraphs

A colouring of a digraph as defined by Erdos and Neumann-Lara in 1980 is...
research
09/13/2022

TDB: Breaking All Hop-Constrained Cycles in Billion-Scale Directed Graphs

The Feedback vertex set with the minimum size is one of Karp's 21 NP-com...
research
12/17/2019

Erdős-Pósa from ball packing

A classic theorem of Erdős and Pósa (1965) states that every graph has e...
research
07/01/2022

Packing cycles in planar and bounded-genus graphs

We devise constant-factor approximation algorithms for finding as many d...
research
08/30/2023

Data reduction for directed feedback vertex set on graphs without long induced cycles

We study reduction rules for Directed Feedback Vertex Set (DFVS) on inst...

Please sign up or login with your details

Forgot password? Click here to reset