Abstract The goal of a fugitive-search game on a graph is to trap a fugitive that hides on the vertices of the graph by systematically placing searchers on the vertices. The fugitive is assumed to have complete knowledge of the graph and of the searchers' moves, but is restricted to move only along paths whose vertices are not guarded by searchers. The search number of the graph is the least number of searchers necessary to trap the fugitive. Variants of the fugitive-search game have been related to important graph parameters like treewidth and pathwidth. In this paper, we introduce a class of fugitive-search games where the searchers do not see the fugitive and the fugitive can only move just before a searcher is placed on the vertex it occupies. Letting the fugitive's speed (i.e. the maximum number of edges the fugitive can traverse at a single move) vary, we get different games. We show that if the speed of the fugitive is unbounded then the search number minus 1 is equal to the treewidth of the graph, while if the speed is 1 then the search number minus 1 is equal to a polynomially computable graph parameter which is called width, or alternatively linkage, and is studied in the context of the Constraint Satisfaction and Boolean Satisfiability Problems. We also show that in the above two cases, the search number remains the same even if we consider only search strategies that at every step further restrict the fugitive's possible resorts (this monotonicity property is usually expressed as: “recontamination does not help”). Moreover, we give an equivalent characterization of the search number for any given fugitive speed in terms of an elimination ordering of the vertices of the graph. Using this characterization, we show that for any graph, if the length of any chordless cycle is bounded by a constant s ( s ⩾ 3), then the treewidth of the graph plus 1 is equal to the search number for fugitive speed s − 2.