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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.
This states that families of graphs closed under the graph minor operation may be characterized by a finite set of forbidden minors. As part of this work, Robertson and Seymour also proved the graph structure theorem describing the graphs in these families. [6] Additional major results in Robertson's research include the following:
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 ...
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By Robertson–Seymour theorem, any set of finite graphs contains only a finite number of minor-minimal elements. In particular, the set of "yes" instances has a finite number of minor-minimal elements. Given an input graph G, the following "algorithm" solves the above problem: For every minor-minimal element H: If H is a minor of G then return ...