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The Robertson–Seymour theorem states that finite undirected graphs and graph minors form a well-quasi-ordering. The graph minor relationship does not contain any infinite descending chain, because each contraction or deletion reduces the number of edges and vertices of the graph (a non-negative integer). [8]
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.
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 ...
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.
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 ...
Finite graphs ordered by a notion of embedding called "graph minor" is a well-quasi-order (Robertson–Seymour theorem). Graphs of finite tree-depth ordered by the induced subgraph relation form a well-quasi-order, [3] as do the cographs ordered by induced subgraphs. [4]
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.
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: