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A similar theorem states that K 4 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.
The non-constructive part here is the Robertson–Seymour theorem. Although it guarantees that there is a finite number of minor-minimal elements it does not tell us what these elements are. Therefore, we cannot really execute the "algorithm" mentioned above. But, we do know that an algorithm exists and that its runtime is polynomial.
The Robertson–Seymour theorem proves that subcubic graphs (simple or not) are well-founded by homeomorphic embeddability, implying such a sequence cannot be infinite. Then, by applying Kőnig's lemma on the tree of such sequences under extension, for each value of k there is a sequence with maximal length.
Symbolab is an answer engine [1] that provides step-by-step solutions to mathematical problems in a range of subjects. [2] It was originally developed by Israeli start-up company EqsQuest Ltd., under whom it was released for public use in 2011.
The Robertson–Seymour theorem implies that a minor-hereditary property may be characterized in terms of a finite set of forbidden minors. The term "hereditary" has been also used for graph properties that are closed with respect to taking subgraphs. [3]
Some problems are known to be solvable in polynomial time, but no concrete algorithm is known for solving them. For example, the Robertson–Seymour theorem guarantees that there is a finite list of forbidden minors that characterizes (for example) the set of graphs that can be embedded on a torus; moreover, Robertson and Seymour showed that ...
1974 4-color theorem. Appel and Haken's proof of this took 139 pages, and also depended on long computer calculations. 1974 The Gorenstein–Harada theorem classifying finite groups of sectional 2-rank at most 4 was 464 pages long. 1976 Eisenstein series. Langlands's proof of the functional equation for Eisenstein series was 337 pages long.
This case of the theorem is still provable by Π 1 1-CA 0, but by adding a "gap condition" [3] to the definition of the order on trees above, he found a natural variation of the theorem unprovable in this system. [4] [5] Much later, the Robertson–Seymour theorem would give another theorem unprovable by Π 1 1-CA 0.