rayless (that is, having no ends). Havens are also closely related to the existence of separators, small sets of vertices in an -vertex graph such that every -flap has at most vertices. If a graph does not have a -vertex separator, then every set of at most vertices has a (unique) -flap with more than vertices. In this case, has a haven of order , in which is defined to be this unique large -flap. That is, every graph has either a small separator or a haven of high order. If a graph has a haven of order , with for some integer , then must also have a complete graph as a minor. In other words, the Hadwiger number of an -vertex graph with a haven of order is at least . As a consequence, the -minor-free graphs have treewidth less than and separators of size less than . More generally an bound on treewidth and separator size holds for any nontrivial family of graphs that can be characterized by forbidden minors, because for any such family there is a constant such that the family does not include . In infinite graphs If a graph contains a ray, a semi-infinite simple path with a starting vertex but no ending vertex, then it has a haven of order : that is, a function that maps each finite set of vertices to an -flap, satisfying the consistency condition for havens. Namely, define to be the unique -flap that contains infinitely many vertices of the ray. Thus, in the case of infinite graphs the connection between treewidth and havens breaks down: a single ray, despite itself being a tree, has havens of all finite orders and even more strongly a haven of order . Two rays of an infinite graph are considered to be equivalent if there is no finite set of vertices that separates infinitely many vertices of one ray from infinitely many vertices of the other ray; this is an equivalence relation, and its equivalence classes are called ends of the graph. The ends of any graph are in one-to-one correspondence with its havens of order . For, every ray determines a haven, and every two equivalent rays determine the same haven. Conversely, every haven is determined by a ray in this way, as can be shown by the following case analysis: If the haven has the property that the