Monotonicity and inert fugitive search games

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 “clearing” an edge during a search: when a sliding of a searcher occurs along the edge or when both its endpoints are simultaneously occupied by searchers. These variants define the inert edge search and the inert node search respectively. A third search variant, the inert mixed search, is defined when both ways of clearing an edge are possible. As we show, inert search and inert mixed search are equivalent (surprisingly, this is not the case if we discard the inertness property). Moreover, we prove that, in any case, by restricting the searches to only those that always reduce further the fugitive’s possible resorts, does not give any advantage to the fugitive (this monotonicity property is usually expressed as: “recontamination does not help”). So far, the only monotonicity result on inert search concerns inert node search and our results yield a much simpler proof of that result as well. Furthermore, we define a new graph-theoretic parameter, the proper-treewidth, in analogy to the parameter proper-pathwidth, and prove it equivalent to the inert mixed search game. Last, we prove that proper-treewidth, in turn, is equivalent to a known graph theoretic parameter related to treewidth. Preprint submitted to Elsevier Preprint 20 December 2007