Directed path-decompositions

TL;DR

This paper extends tools from tree-decomposition theory to directed graphs with a focus on directed path-width, introducing new concepts like directed blockage and proving related min-max theorems and structural properties.

Contribution

It introduces the notion of directed blockage, establishes a min-max theorem for directed path-width, and shows every digraph admits a linked directed path-decomposition of minimum width.

Findings

Established a min-max theorem for directed path-width.

Proved every digraph with large directed path-width contains certain arborescences as butterfly minors.

Showed existence of linked directed path-decompositions of minimum width.

Abstract

Many of the tools developed for the theory of tree-decompositions of graphs do not work for directed graphs. In this paper we show that some of the most basic tools do work in the case where the model digraph is a directed path. Using these tools we define a notion of a directed blockage in a digraph and prove a min-max theorem for directed path-width analogous to the result of Bienstock, Roberston, Seymour and Thomas for blockages in graphs. Furthermore, we show that every digraph with directed path width $\geq k$ contains each arboresence of order $\leq k + 1$ as a butterfly minor. Finally we also show that every digraph admits a linked directed path-decomposition of minimum width, extending a result of Kim and Seymour on semi-complete digraphs.