Characterizing DAG-depth of Directed Graphs

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.