Edge searching weighted graphs

In traditional edge searching one tries to clean all of the edges in a graph employing the least number of searchers. It is assumed that each edge of the graph initially has a weight equal to one. In this paper we modify the problem and introduce the Weighted Edge Searching Problem by considering graphswith arbitrary positive integerweights assigned to its edges. We give bounds on the weighted search number in terms of related graph parameters including pathwidth. We characterize the graphs for which two searchers are sufficient to clear all edges. We show that for every weighted graph the minimum number of searchers needed for a not-necessarily-monotonic weighted edge search strategy is enough for a monotonicweighted edge search strategy,where each edge is cleaned only once. This result proves the NP-completeness of the problem. Crown Copyright© 2008 Published by Elsevier B.V. All rights reserved.