Characterizing DAG-depth of Directed Graphs

TL;DR

This paper introduces DAG-depth, a measure of directed graph complexity, and develops a strategy-based decomposition method for the cops-and-robber game, extending concepts from tree-depth to directed graphs.

Contribution

It defines DAG-depth decomposition, proves its correctness, and offers methods to optimize and compute the decomposition for strategic game play.

Findings

DAG-depth extends tree-depth to directed graphs.

A correct DAG-depth decomposition strategy is established.

Methods for optimizing and computing the decomposition are provided.

Abstract

We study DAG-depth, a structural depth measure of directed graphs, which naturally extends the tree-depth of ordinary graphs. We define a DAG-depth decomposition as a strategy for the cop player in the lift-free version of the cops-and-robber game on directed graphs and prove its correctness. The DAG-depth decomposition is related to DAG-depth in a similar way as an elimination tree is related to the tree-depth. We study the size aspect of DAG-depth decomposition and provide a definition of mergeable and optimally mergeable vertices in order to make the decomposition smaller and acceptable for the cop player as a strategy. We also provide a way to find the closure of a DAG-depth decomposition, which is the largest digraph for which the given decomposition represents a winning strategy for the cop player.