Edge searching and fast searching with constraints

In this paper, we consider the edge searching problem and the fast searching problem with constraints on some vertices. The edge searching problem was introduced by Megiddo et al. (1988) [15], in which a group of searchers want to capture an invisible robber. The fast searching problem was introduced by Dyer et al. (2008) [8]. One constraint we consider for these two searching models is that a subset of vertices, called start vertices, are initially occupied by searchers. We want to find the minimum number of additional searchers to capture the robber. Another constraint is that there are some vertices, called halt vertices, which have to be occupied by searchers at the end of the searching process. For the edge searching on a tree containing only start vertices, we propose an O(nlog⁡n)-time algorithm for computing the edge search number, where n is the number of vertices. For the edge searching on a tree with start vertices and halt vertices, we give an O(n²)-time algorithm to compute the edge search number. For the fast searching on a tree that contains only start vertices, we present a linear-time algorithm for computing the fast search number. For the fast searching on a tree with n vertices that contains s start vertices and h halt vertices, we propose an O((s+h)n)-time algorithm to compute the fast search number. For the edge searching (respectively, fast searching) with only halt vertices, we reduce the problem to the edge searching (respectively, fast searching) with start vertices. © 2024 Elsevier B.V.