A Note on Directed Treewidth

TL;DR

This paper characterizes digraphs with directed treewidth one using forbidden minors and establishes a linear relation between hypertree-width of cycle hypergraphs and directed treewidth, linking structural properties to hypergraph theory.

Contribution

It provides a forbidden minor characterization for directed treewidth one and relates it to hypertree-width of cycle hypergraphs, offering new structural insights.

Findings

Directed treewidth one characterized by forbidden butterfly minors

Linear relation between hypertree-width of cycle hypergraph and directed treewidth

Digraphs with directed treewidth one have hypertree cycle hypergraphs

Abstract

We characterise digraphs of directed treewidth one in terms of forbidden butterfly minors. Moreover, we show that there is a linear relation between the hypertree-width of the dual of the cycle hypergraph of D, i. e. the hypergraph with vertices V (D) where every hyperedge corresponds to a directed cycle in D, and the directed treewidth of D. Based on this we show that a digraph has directed treewidth one if and only if its cycle hypergraph is a hypertree.