This is a short post about the following paper in my directed minor series :

- Shiva Kintali. “
**Directed Minors III. Directed Linked Decompositions**“. Preprint available on my publications page.

Thomas [Tho'90] proved that every undirected graph admits a linked tree decomposition of width equal to its treewidth. This theorem is a key technical tool for proving that every set of bounded treewidth graphs is well-quasi-ordered. An analogous theorem for branch-width was proved by Geelen, Gerards and Whittle [GGW'02]. They used this result to prove that all matroids representable over a fixed finite field and with bounded branch-width are well-quasi-ordered under minors. Kim and Seymour [KS'12] proved that every semi-complete digraph admits a linked directed path decomposition of width equal to its directed pathwidth. They used this result to show that all semi-complete digraphs are well-quasi-ordered under “strong” minors.

In this paper, we generalize Thomas’s theorem to all digraphs.

Theorem: Every digraph G admits a linked directed path decomposition and a linked DAG decomposition of width equal to its directed pathwidth and DAG-width respectively.

The above theorem is crucial to prove well-quasi-ordering of some interesting classes of digraphs. I will release Directed Minors IV soon. Stay tuned !!