Analogue of the Excluded Grid Theorem for acyclic digraphs

Determine whether there exists an analogue of the Robertson–Seymour Excluded Grid Theorem for acyclic digraphs that uses the k×k acyclic directed grid (with all horizontal paths directed left-to-right) as the grid structure, i.e., establish whether a width–grid equivalence theorem holds for acyclic digraphs analogous to the undirected case.

Background

In undirected graphs, the Robertson–Seymour Excluded Grid Theorem (GM5) states a width–grid equivalence: large tree-width forces a large grid minor, and conversely a large grid minor implies large tree-width. Extending such results to digraphs requires specifying appropriate notions of directed grid and minor containment.

The paper discusses multiple directed grid models (acyclic grid, semi-grid, alternating grid) and shows close relationships among some of them under embedding. For semi-grids and cylindrical grids, grid theorems analogous to the undirected case are known via the directed tree-width framework of Kawarabayashi and Kreutzer. However, the authors explicitly note that no such analogue is currently known for acyclic grids, raising the question of whether a directed excluded grid theorem can be formulated for acyclic digraphs.