Jumping robbers in digraphs

We generalise the results of Richerby and Thilikos in two aspects. We give a linear bound on the number of additional cops in the game (1) on directed graphs and (2) where the robbers have more power: every robber can leave his vertex and jump to another robber. We show that our bound is tight and that the reason for a worse lower bound is not the directedness of the graphs, but the ability of the robbers to jump. We also prove that adding new robbers induces a hierarchy of the cop numbers for a fixed graph which is strict on some graphs and converges to the minimum number of cops needed to capture an invisible robber (as in [25]).