Towards the Graph Minor Theorems for Directed Graphs

Abstract

Two key results of Robertson and Seymour’s graph minor theory are:

• 1. a structure theorem stating that all graphs excluding some fixed graph as a minor have a tree decomposition into pieces that are almost embeddable in a fixed surface.

• 2. the k-disjoint paths problem is tractable when $k$ is a fixed constant: given a graph $G$ and $k$ pairs $(s_1,t_1)$, ..., $(s_k,t_k)$ of vertices of $G$, decide whether there are $k$ mutually vertex disjoint paths of $G$, the $i$th path linking $s_i$ and $t_i$ for $i=1,\dots ,k$.

In this talk, we shall try to look at the corresponding problems for digraphs.

Concerning the first point, the grid theorem, originally proved in 1986 by Robertson and Seymour in Graph Minors V, is the basis (even for the whole graph minor project). In the mid-90s, Reed and Johnson, Robertson, Seymour and Thomas (see [13, 26]), independently, conjectured an analogous theorem for directed graphs, i.e. the existence of a function $f\mathrel : {\mathbb {N}} \rightarrow {\mathbb {N}}$ such that every digraph of directed treewidth at least $f(k)$ contains a directed grid of order $k$. In an unpublished manuscript from 2001, Johnson, Robertson, Seymour and Thomas give a proof of this conjecture for planar digraphs. But for over a decade, this was the most general case proved for the conjecture.

We are finally able to confirm the Reed, Johnson, Robertson, Seymour and Thomas conjecture in full generality. As a consequence of our results we are able to improve results in Reed et al. in 1996 [27] to disjoint cycles of length at least $l$. This would be the first but a significant step toward the structural goals for digraphs (hence towards the first point).

Concerning the second point, in [19] we contribute to the disjoint paths problem using the directed grid theorem. We show that the following can be done in polynomial time:

Suppose that we are given a digraph $G$ and $k$ terminal pairs $(s_1, t_1), (s_2, t_2), \dots , (s_k, t_k)$, where $k$ is a fixed constant. In polynomial time, either

• we can find $k$ paths $P_1, \dots , P_k$ such that $P_i$ is from $s_i$ to $t_i$ for $i=1, \dots , k$ and every vertex in $G$ is in at most four of the paths, or

• we can conclude that $G$ does not contain disjoint paths $P_1, \dots , P_k$ such that $P_i$ is from $s_i$ to $t_i$ for $i=1, \dots , k$.

To the best of our knowledge, this is the first positive result for the general directed disjoint paths problem (and hence for the second point). Note that the directed disjoint paths problem is NP-hard even for $k=2$. Therefore, this kind of results is the best one can hope for.

We also report some progress on the above two points.

Ken-ichi Kawarabayashi—This work was supported by JST ERATO Kawarabayashi Large Graph Project and by Mitsubishi Foundation.

Stephan Kreutzer—This project has received funding from the European Research Council (ERC) under the European Unions Horizon 2020 research and innovation programme (grant agreement No 648527).