The complexity of searching a graph

T. Parsons originally proposed and studied the following pursuit-evasion problem on graphs: Members of a team of searchers traverse the edges of a graph G in pursuit of a fugitive, who moves along the edges of the graph with complete knowledge of the locations of the pursuers. What is the smallest number s ( G ) of searchers that will suffice for guaranteeing capture of the fugitive? It is shown that determining whether s ( G ) ≤ K , for a given integer K , is NP-complete for general graphs but can be solved in linear time for trees. We also provide a structural characterization of those graphs G with s ( G ) ≤ K for K = 1, 2, 3.