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 $kâ‰\yen 2$, 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.

中文翻译：

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连接搜索一个懒惰的强盗

针对懒惰（或分别为敏捷）隐形强盗的节点搜索游戏已被引入作为树宽参数（分别为路径宽度）的搜索游戏模拟。在上述游戏的连接变体中，我们还要求在搜索的每个时刻都连接干净的区域。针对敏捷且隐形的强盗的连接搜索游戏已被广泛研究。这个游戏的单调变体（我们要求干净的区域逐渐增加）对应于连接路径宽度的图参数。众所周知，连通性的代价寻找敏捷强盗的边界为 2，即图的连通路径宽度至多是其路径宽度的两倍（加上某个常数）。我们调查了针对懒惰强盗的连接搜索游戏的研究。一个懒惰的强盗只有在警察的策略威胁到他目前占据的顶点时才会行动。我们介绍了这个游戏的两种替代图论公式，一种是连接树分解，一种是（连接）布局，导致连接树宽度的图参数。我们观察到连接的树宽参数在收缩下是封闭的，并证明对于每个$克\ge 2$, 最大连通树宽的图类的收缩障碍集 $克$是无限的。我们的主要结果是对障碍集的完整表征$克=2$. 我们还表明，与敏捷强盗游戏相比，连接的代价是无限的。