moving, the evader may not pass through any vertex that is already occupied by any of the pursuers. If a -haven (with the monotonicity property) exists, then the evader may avoid being captured indefinitely, and win the game, by always moving to a vertex of where is the set of vertices that will be occupied by pursuers at the end of the move. The monotonicity property of a haven guarantees that, when a new pursuer is added to a vertex of the graph, the vertices in are always reachable from the current position of the evader. For instance, an evader can win this game against three pursuers on a grid by following this strategy with the haven of order 4 described in the example. However, on the same graph, four pursuers can always capture the evader, by first moving onto three vertices that split the grid onto two three-vertex paths, then moving into the center of the path containing the evader, forcing the evader into one of the corner vertices, and finally removing one of the pursuers that is not adjacent to this corner and placing it onto the evader. Therefore, the grid can have no haven of order 5. Havens with the touching property allow the evader to win the game against more powerful pursuers that may simultaneously jump from one set of occupied vertices to another. Connections to treewidth, separators, and minors Havens may be used to characterize the treewidth of graphs: a graph has a haven of order if and only if it has treewidth at least . A tree decomposition may be used to describe a winning strategy for the pursuers in the same pursuit–evasion game, so it is also true that a graph has a haven of order if and only if the evader wins with best play against fewer than pursuers. In games won by the evader, there is always an optimal strategy in the form described by a haven, and in games won by the pursuer, there is always an optimal strategy in the form described by a tree decomposition. For instance, because the grid has a haven of order 4, but does not have a haven of order 5, it must have treewidth exactly 3. The same min-max theorem can be generalized to infinite graphs of finite treewidth, with a definition of treewidth in which the underlying tree is required to be