Deciding the Erdős-Pósa Property in 3-Connected Digraphs

Abstract

A (di)graph H has the Erdős-Pósa (EP) property for (butterfly) minors if there exists a function $f: \mathbb {N} \rightarrow \mathbb {N}$ such that, for any $k\in \mathbb {N}$ and any (di)graph G, either G contains at least k pairwise vertex-disjoint copies of H as (butterfly) minor, or there exists a subset T of at most f(k) vertices such that H is not a (butterfly) minor of $G-T$. It is a well known result of Robertson and Seymour that an undirected graph has the EP property if and only if it is planar. This result was transposed to digraphs by Amiri, Kawarabayashi, Kreutzer and Wollan, who proved that a strong digraph has the EP property for butterfly minors if, and only if, it is a butterfly minor of a cylindrical grid. Contrary to the undirected case where a graph is planar if, and only if, it is the minor of some grid, not all planar digraphs are butterfly minors of a cylindrical grid. In this work, we characterize the planar digraphs that have a butterfly model in a cylindrical grid. In particular, this leads to a linear-time algorithm that decides whether a weakly 3-connected strong digraph has the EP property.

(Partially) supported by: FUNCAP MLC-0191-00056.01.00/22 and PNE-0112-00061.01.00/16, and CNPq 303803/2020-7, the CAPES-Cofecub project Ma 1004/23, by the project UCA JEDI (ANR-15-IDEX-01) and EUR DS4H Investments in the Future (ANR-17-EURE-004), the ANR Digraphs, the ANR Multimod and the Inria Associated Team CANOE. The full versions of omitted or sketched proofs can be found in [3].