In general, graph search can be described in terms of a sequence of searchers' moves on a graph, able to capture a fugitive resorting on its vertices/edges. Several variations of graph search have been introduced, differing on the abilities of the fugitive as well as of the search. In this paper, we examine the case where the fugitive is inert, i.e., it moves only whenever the search is about to capture it. Mainly, there are two variants for