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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.
A minor of a graph G is any graph H that is isomorphic to a graph that can be obtained from a subgraph of G by contracting some edges. If G does not have a graph H as a minor, then we say that G is H-free. Let H be a fixed graph. Intuitively, if G is a huge H-free graph, then there ought to be a "good reason" for this.
Another result relating the four-color theorem to graph minors is the snark theorem announced by Robertson, Sanders, Seymour, and Thomas, a strengthening of the four-color theorem conjectured by W. T. Tutte and stating that any bridgeless 3-regular graph that requires four colors in an edge coloring must have the Petersen graph as a minor.
Since every minor of a planar graph is itself planar, this gives a planar cover of the minor G. Because the graphs with planar covers are closed under the operation of taking minors, it follows from the Robertson–Seymour theorem that they may be characterized by a finite set of forbidden minors. [7] A graph is a forbidden minor for this ...
The set of forbidden minors for the linklessly embeddable graphs was identified by Sachs (1983): the seven graphs of the Petersen family are all minor-minimal intrinsically linked graphs. However, Sachs was unable to prove that these were the only minimal linked graphs, and this was finally accomplished by Robertson, Seymour & Thomas (1995).
Pages in category "Graph minor theory" The following 33 pages are in this category, out of 33 total. ... Robertson–Seymour theorem; S. Shallow minor; Snark (graph ...
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Suppose we have a sequence of simple subcubic graphs G 1, G 2, ... such that each graph G i has at most i + k vertices (for some integer k) and for no i < j is G i homeomorphically embeddable into (i.e. is a graph minor of) G j. The Robertson–Seymour theorem proves that subcubic graphs (simple or not) are well-founded by homeomorphic ...