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. © 1988, ACM. All rights reserved.