Fast edge searching and fast searching on graphs

Given a graph G=(V,E) in which a fugitive hides on vertices or along edges, graph searching problems are usually to find the minimum number of searchers required to capture the fugitive. In this paper, we consider the problem of finding the minimum number of steps to capture the fugitive. We introduce the fast edge searching problem in the edge search model, which is the problem of finding the minimum number of steps (called the fast edge-search time) to capture the fugitive. We establish relations between the fast edge searching and the fast searching that is the problem of finding the minimum number of searchers to capture the fugitive in the fast search model. While the family of graphs whose fast search number is at most k is not minor-closed for any positive integer k>=2, we show that the family of graphs whose fast edge-search time is at most k is minor-closed. We establish relations between the fast (fast edge) searching and the node searching. These relations allow us to transform the problem of computing node search numbers to the problem of computing fast edge-search numbers or fast search numbers. Using these relations, we prove that the problem of deciding, given a graph G and an integer k, whether the fast (edge-)search number of G is less than or equal to k is NP-complete; and it remains NP-complete for Eulerian graphs. We also prove that the problem of determining whether the fast (edge-)search number of G is half of the number of odd vertices in G is NP-complete; and it remains NP-complete for planar graphs with maximum degree 4. We present a linear time approximation algorithm for the fast edge-search time that always delivers solutions of at most (1+|V|-1|E|+1) times the optimal value. This algorithm also gives us a tight upper bound on the fast search number of graphs. We also show a lower bound on the fast search number using the minimum degree and the number of odd vertices.