On the D-Width of Directed Graphs

The cops and strong visible robber game, defined in [3], is played according to the following rules: Initially, the cops occupy some set of X ⊆ V vertices, with |X| ≤ k, and the robber is placed on some vertex v ∈ V − X. At any time, some of the cops can reside outside the graph, say, in a helicopter. In each round, the cop player chooses the next location X ′ ⊆ V for the cops. The stationary cops in X ∩ X ′ remain in their positions, while the others go to the helicopter and fly to their new position. During this, the robber player, knowing the cops' next position X ′ from wiretapping the police radio, can run at great speed to any new position v ′ , provided there is both a (possibly empty) directed path from v to v ′ , and a (possibly empty) directed path back from v ′ to v in G − (X ∩ X ′), that is, he has to avoid to run into a stationary cop, and to run along a path in and to stay in the same strongly connected component of the remaining graph induced by the non-blocked vertices. Afterwards, the helicopter lands the cops at their new positions, and the next round starts, with X ′ and v ′ taking over the roles of X and v, respectively. The cop player wins the game if eventually a cop lands on the robber's current position or robber cannot move any more, and the robber player wins if the robber can escape indefinitely. Instead of letting the robber choose a single vertex in each turn, we prefer to think of the robber's positions as strongly connected components of G, and of the robber choosing in each turn a strongly connected component R