Forbidden Directed Minors and Kelly-width

TL;DR

This paper extends the characterization of partial 1-trees to directed graphs, showing that partial 1-DAGs are characterized by three specific forbidden directed minors, K_3, N_4, and M_5.

Contribution

It introduces a new forbidden minor characterization for partial 1-DAGs, generalizing the undirected case to directed graphs.

Findings

Partial 1-DAGs are characterized by three forbidden directed minors.

The characterization generalizes the undirected case of partial 1-trees.

Provides a foundation for understanding directed graph minors and width parameters.

Abstract

Partial 1-trees are undirected graphs of treewidth at most one. Similarly, partial 1-DAGs are directed graphs of KellyWidth at most two. It is well-known that an undirected graph is a partial 1-tree if and only if it has no K_3 minor. In this paper, we generalize this characterization to partial 1-DAGs. We show that partial 1-DAGs are characterized by three forbidden directed minors, K_3, N_4 and M_5.