Haven (graph Theory) - Definition

Definition

Formally, if G is an undirected graph, and X is a set of vertices, then an X-flap is a nonempty connected component of the subgraph of G formed by deleting X. A haven of order k in G is a function β assigning an X-flap β(X) to every set X of fewer than k vertices, satisfying additional constraints which are given differently by different authors. The number k 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 β(X) and β(Y) 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 the following monotonicity property: if X ⊆ Y, and both X and Y have fewer than k vertices, then β(Y) ⊆ β(X). 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 β(X) for a haven of order k (with the touching definition) forms a bramble of order k. Conversely, given a bramble of order k, one may define a haven of the same order, by defining β(X) to be the X-flap that includes all of the subgraphs in the bramble that are disjoint from X. The requirement that the subgraphs in the bramble all touch each other ensures that this X-flap is unique, and that all chosen flaps touch each other.