  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.
