Robertson–Seymour Theorem - Obstruction Sets

Obstruction Sets

Some examples of finite obstruction sets were already known for specific classes of graphs before the Robertson–Seymour theorem was proved. For example, the obstruction for the set of all forests is the loop graph (or, if one restricts to simple graphs, the cycle with three vertices). This means that a graph is a forest if and only if none of its minors is the loop (or, the cycle with three vertices, respectively). The sole obstruction for the set of paths is the tree with four vertices, one of which has degree 3. In these cases, the obstruction set contains a single element, but in general this is not the case. Wagner's theorem states that a graph is planar if and only if it has neither K₅ nor K_3,3 as a minor. In other words, the set {K₅, K_3,3} is an obstruction set for the set of all planar graphs, and in fact the unique minimal obstruction set. A similar theorem states that K₄ and K_2,3 are the forbidden minors for the set of outerplanar graphs.

Although the Robertson–Seymour theorem extends these results to arbitrary minor-closed graph families, it is not a complete substitute for these results, because it does not provide an explicit description of the obstruction set for any family. For example, it tells us that the set of toroidal graphs has a finite obstruction set, but it does not provide any such set. The complete set of forbidden minors for toroidal graphs remains unknown, but contains at least 16000 graphs.