The excluded grid theorem, originally proved by Robertson and Seymour in Graph Minors V, is one of the most central results in the study of graph minors. It has found numerous applications in algorithmic graph structure theory, for instance as the basis for bidimensionality theory on graph classes excluding a fixed minor. In 1997, Reed [22] and later Johnson, Robertson, Seymour and Thomas [16] conjectured an analogous theorem for directed graphs, i.e. the existence of a function f : N → N such that every digraph of directed tree-width at least f(k) contains a directed grid of order k. In an unpublished manuscript from 2001, Johnson, Robertson, Seymour and Thomas gave a proof of this conjecture for planar digraphs but no result beyond planar graphs is known to date. In this paper we prove the conjecture for the case of digraphs excluding a fixed undirected graph as a mi- nor. For algorithmic applications our theorem is particularly interesting as it covers those classes of digraphs to which, on undirected graphs, theories based on the excluded grid theorem such as bidimensionality theory apply. We expect similar applications for directed graphs in particular to algorithmic versions of Erdos-Posa type results and the directed disjoint paths problem.
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