Maximum vertex occupation time and inert fugitive: Recontamination does help

Given a simple graph G, we consider the node search problem with inert fugitive. We are interested in minimizing the maximum vertex occupation time, i.e. the maximum number of steps in which a vertex is occupied by a searcher during a search of G. We prove that a search program which does not allow a recontamination may not find an optimal solution to this problem. Moreover, the difference between the minimum maximum vertex occupation time computed by a monotone search program and a program without such a restriction may be arbitrarily large.