Connected Search for a Lazy Robber

The node-search game against a lazy (or, respectively, agile) invisible robber has been introduced as a search-game analogue of the treewidth parameter (and, respectively, pathwidth). In the connected variants of the above games, we additionally demand that, at each moment of the search, the clean territories are connected. The connected search game against an agile and invisible robber has been extensively examined. The monotone variant (where we demand that the clean territories are progressively increasing) of this game corresponds to the graph parameter of connected pathwidth. It is known that the price of connectivity to search for an agile robber is bounded by 2, that is, the connected pathwidth of a graph is at most twice (plus some constant) its pathwidth. We investigate the study of the connected search game against a lazy robber. A lazy robber moves only when the cops' strategy threatens the vertex that he currently occupies. We introduce two alternative graph-theoretical formulations of this game, one in terms of connected tree-decompositions and one in terms of (connected) layouts, leading to the graph parameter of connected treewidth. We observe that the connected treewidth parameter is closed under contractions and prove that for every k ≥ 2 , the set of contraction obstructions of the class of graphs with connected treewidth at most k is infinite. Our main result is a complete characterization of the obstruction set for k = 2 . We also show that, in contrast to the agile robber game, the price of connectivity is unbounded.