  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, and the
difference between the maximum vertex occupation time computed by a monotone
search program and a program without such restriction may be arbitrarily large.
