Commmm~mml by A,S+ Fn~kd A robber and k cop~ chno~ starting veaices in a graph and move in alternation from vortex to vertex along the edges of the graph; captttre occurs if a cop ever sha~ a vertex with the robber, if k is not fixed and if either the graph is directed or initial portions are given, we show that the pmblern is EXPTIME-cxnnplete. Similar techniques lead to the PSPACE-and EXPTIME~.ompleteness of ~ other combi~torial gaines that were Weviously only known to be NP-and PSPACE-berd. A robber and k cops choose starting vertices in a graph and move in alternation from vertex to vertex along the edges of the graph; all know the locations of the others and can choose to remain stationary rather than move. Capture occurs if a cop and the robber share the same vertex -the cops try to capture the robber who tries to avoid capture, The case of k = 1 on an undit~;ted graph was iti~ and studied by QoiUiot [25, pp, 131-145; 26,28] and by Nowaknwski and WinHer [22]. A simple algorithm deXermines if one cop can capture a robber on a given graph: eonwaet the graph by repeatedly removing vertices v for which there exists an adjacent vertex u such that all other vertices adjacent to p are also adjacent to u; that is, u covers v and all of v*s neighbors, If the algorithm terminates with one vertex remaining then the cop can capture the robber, otherwise the robber can evade qutpture.
The problem of determining whether k cops with given initial locations can capture a robber on a given undirected graph is EXPTIME-complete REF .
The complexity of pursuit on a graph
Theor. Comput. Sci.
Theor. Comput. Sci.
Computer Science