In graph theory, a haven is a certain type of function on sets of vertices in an undirected graph. If a haven exists, it can be used by an evader to win a pursuit–evasion game on the graph, by consulting the function at each step of the game to determine a safe set of vertices to move into. Havens were first introduced by as a tool for characterizing the treewidth of graphs. Their other applications include proving the existence of small separators on minor-closed families of graphs, and characterizing the ends and clique minors of infinite graphs. Definition If is an undirected graph, and is a set of vertices, then an -flap is a nonempty connected component of the subgraph of formed by deleting . A haven of order in is a function that assigns an -flap to every set of fewer than vertices. This function must also satisfy additional constraints which are given differently by different authors. The number is called the order of the haven. In the original definition of Seymour and Thomas, a haven is required to satisfy the property that every two flaps and must touch each other: either they share a common vertex or there exists an edge with one endpoint in each flap. In the definition used later by Alon, Seymour, and Thomas, havens are instead required to satisfy a weaker monotonicity property: if , and both and have fewer than vertices, then . The touching property implies the monotonicity property, but not necessarily vice versa. However, it follows from the results of Seymour and Thomas that, in finite graphs, if a haven with the monotonicity property exists, then one with the same order and the touching property also exists. Havens with the touching definition are closely related to brambles, families of connected subgraphs of a given graph that all touch each other. The order of a bramble is the minimum number of vertices needed in a set of vertices that hits all of the subgraphs in the family. The set of flaps for a haven of order (with the touching definition) forms a bramble of order at least , because any set of fewer than vertices fails to hit the subgraph . Conversely, from any bramble of order , one may construct a haven of the same order, by defining (for each choice of ) to be the -flap that includes all of