Given a graph that contains an invisible fugitive, the fast searching problem is to find the fast search number, i.e., the minimum number of searchers to capture the fugitive in the fast search model. In this paper, we give a new lower bound on the fast search number. Using the new lower bound, we prove an explicit formula for the fast search number of the Cartesian product of an Eulerian graph and a path. We also give formulas for the fast search number of variants of the Cartesian product. We present an upper bound and a lower bound on the fast search number of hypercubes, and extend the results to a broader class of graphs including toroidal grids.
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