# Forbidden Directed Minors and Directed Pathwidth

Today’s post is about the following paper, a joint work with Qiuyi Zhang, one of my advisees. Qiuyi Zhang is now a graduate student in the Mathematics department of Berkeley.

• Shiva Kintali, Qiuyi Zhang. Forbidden Directed Minors and Directed Pathwidth. (Preprint is available on my publications page)

Undirected graphs of pathwidth at most one are characterized by two forbidden minors i.e., (i) $K_3$ the complete graph on three vertices and (ii) $S_{2,2,2}$ the spider graph with three legs of length two each (see the following figure).

Directed pathwidth is a natural generalization of pathwidth to digraphs. We proved that digraphs of directed pathwidth at most one are characterized by a finite number of forbidden directed minors. In particular, we proved that the number of vertices in any forbidden directed minor is at most 8*160000+7. Ahem !!

This paper falls in the “directed minors” part of my research interests. In an earlier theorem, proved in April 2013 (see this earlier post), we proved that partial 1-DAGs are characterized by three forbidden directed minors. In a similar vein, I conjectured that the digraphs with directed pathwidth at most 1 are characterized by a finite number of forbidden directed minors. I assumed that the number of forbidden directed minors is number is around 100. So we started this project in May 2013 and started making a list of carefully constructed forbidden directed minors and tried to extend our techniques from partial 1-DAGs. Here is an initial list of minors we found.

All the forbidden minors we found, looked very cute and we assumed that a proof is nigh. Soon, we realized that the list is  growing quickly and none of our earlier techniques are applicable. After almost an year of patient efforts and roller coaster rides, we proved our finiteness theorem in May 2014, two weeks before Qiuyi Zhang’s thesis defense. It took us 10 more months to get the paper to its current status. So this is a two year long adventure.

I am hoping to prove more theorems in the “directed minors” area in the coming years. The current paper taught me that patience and focus are big factors to make consistent progress. There should be a nice balance between `proving new theorems’ and `writing up the existing results’.

# Directed Minors III. Directed Linked Decompositions

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 !!