DAG-width and circumference of digraphs

TL;DR

This paper establishes a relationship between digraph circumference and DAG-width, showing bounded circumference implies bounded DAG-width, and explores the computational complexity of related path and subdigraph problems.

Contribution

It proves that digraphs with bounded circumference have bounded DAG-width and solves the k-linkage problem efficiently for these classes, also analyzing related problems' complexity.

Findings

DAG-width is at most the circumference of the digraph.

k-linkage problem is polynomial for digraphs with bounded circumference.

Minimum spanning strong subdigraph problem is NP-hard on certain digraphs.

Abstract

We prove that every digraph of circumference $l$ has DAG-width at most $l$ and this is best possible. As a consequence of our result we deduce that the $k$-linkage problem is polynomially solvable for every fixed $k$ in the class of digraphs with bounded circumference. This answers a question posed in \cite{bangTCS562}. We also prove that the weak $k$-linkage problem (where we ask for arc-disjoint paths) is polynomially solvable for every fixed $k$ in the class of digraphs with circumference 2 as well as for digraphs with a bounded number of disjoint cycles each of length at least 3. The case of bounded circumference digraphs is open. Finally we prove that the minimum spanning strong subdigraph problem is NP-hard on digraphs of DAG-width at most 5.