Résumé
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 \emph{connected} variants of the above two games, we additionally demand that, at each moment of the search, the \emph{clean} territories are {\sl connected}. The connected search game against an agile and invisible robberhas been extensively examined. The monotone variant (where we also demand that the clean territories are progressively increasing) of this game, corresponds to the graph parameter of {\sl connected pathwidth}. It is known that \emph{the price of connectivtiy} 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.In this paper, we investigate the study of the connected search game against a {\sl lazy} robber. A lazy robber moves only when the searchers' strategy threatens the location 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 {\sl connected treewidth}. We observe that connected treewidth parameter is closed under contractions and prove that for every $k\geq 2$, the set of contraction obstructions of the class of graphs with connected treewidth at most $k$ is infinite.Our main result is a complete characterisation of the obstruction set for $k=2$. One may observe that, so far, only a few complete obstruction sets are explicity known for contraction closed graph classes.We finally show that, in contrast to the agile robber game, the price of connectivity is unbounded.