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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]
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.
A graph H is called a topological minor of a graph G if a subdivision of H is isomorphic to a subgraph of G. [21] Every topological minor is also a minor. The converse however is not true in general (for instance the complete graph K 5 in the Petersen graph is a minor but not a topological one), but holds for graph with maximum degree not ...
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 ...
In 1993, with Seymour and Robin Thomas, Robertson proved the -free case for which the Hadwiger conjecture relating graph coloring to graph minors is known to be true. [ 8 ] In 1996, Robertson, Seymour, Thomas, and Daniel P. Sanders published a new proof of the four color theorem , [ 9 ] confirming the Appel–Haken proof which until then had ...
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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