the subgraphs in the bramble that are disjoint from . The requirement that the subgraphs in the bramble all touch each other can be used to show that this -flap exists, and that all of the flaps chosen in this way touch each other. Thus, a graph has a bramble of order if and only if it has a haven of order . Example As an example, let be a nine-vertex grid graph. Define a haven of order 4 in , mapping each set of three or fewer vertices to an -flap , as follows: If there is a unique -flap that is larger than any of the other -flaps, let be that unique large -flap. Otherwise, choose arbitrarily to be any -flap. It is straightforward to verify by a case analysis that this function satisfies the required monotonicity property of a haven. If and has fewer than two vertices, or has two vertices that are not the two neighbors of a corner vertex of the grid, then there is only one -flap and it contains every -flap. In the remaining case, consists of the two neighbors of a corner vertex and has two -flaps: one consisting of that corner vertex, and another (chosen as ) consisting of the six remaining vertices. No matter which vertex is added to to form , there will be a -flap with at least four vertices, which must be the unique largest flap since it contains more than half of the vertices not in . This large -flap will be chosen as and will be a subset of . Thus in each case monotonicity holds. Pursuit–evasion Havens model a certain class of strategies for an evader in a pursuit–evasion game in which fewer than pursuers attempt to capture a single evader, the pursuers and evader are both restricted to the vertices of a given undirected graph, and the positions of the pursuers and evader are known to both players. At each move of the game, a new pursuer may be added to an arbitrary vertex of the graph (as long as fewer than pursuers are placed on the graph at any time) or one of the already-added pursuers may be removed from the graph. However, before a new pursuer is added, the evader is first informed of its new location and may move along the edges of the graph to any unoccupied vertex. While