Directed Graph Minors and Serial-Parallel Width

TL;DR

This paper introduces new variants of directed graph minors and embeddings, characterizes TDAGs with bounded serial-parallel width, and analyzes the complexity of related computational problems.

Contribution

It proposes novel definitions for directed graph minors, characterizes TDAGs with bounded serial-parallel width, and studies the complexity of related algorithms.

Findings

Directed minors coincide with embeddings for TDAGs.

TDAGs with serial-parallel width 1 are exactly directed series-parallel graphs.

Characterization of all TDAGs with bounded serial-parallel width.

Abstract

Graph minors are a primary tool in understanding the structure of undirected graphs, with many conceptual and algorithmic implications. We propose new variants of \emph{directed graph minors} and \emph{directed graph embeddings}, by modifying familiar definitions. For the class of 2-terminal directed acyclic graphs (TDAGs) our two definitions coincide, and the class is closed under both operations. The usefulness of our directed minor operations is demonstrated by characterizing all TDAGs with serial-parallel width at most $k$; a class of networks known to guarantee bounded negative externality in nonatomic routing games. Our characterization implies that a TDAG has serial-parallel width of $1$ if and only if it is a directed series-parallel graph. We also study the computational complexity of finding a directed minor and computing the serial-parallel width.