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A minor of an undirected graph G is any graph that may be obtained from G by a sequence of zero or more contractions of edges of G and deletions of edges and vertices of G.The minor relationship forms a partial order on the set of all distinct finite undirected graphs, as it obeys the three axioms of partial orders: it is reflexive (every graph is a minor of itself), transitive (a minor of a ...
Perhaps it is true that: for any non-planar graph H, there exists a positive integer k such that every H-free graph can be obtained via k-clique-sums from a list of graphs, each of which either has at most k vertices or embeds on some surface that H does not embed on. Unfortunately, this statement is not yet sophisticated enough to be true.
A graph with n vertices and pathwidth p can be embedded into a three-dimensional grid of size p × p × n in such a way that no two edges (represented as straight line segments between grid points) intersect each other. Thus, graphs of bounded pathwidth have embeddings of this type with linear volume. [54]
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. [15]
An embedded graph uniquely defines cyclic orders of edges incident to the same vertex. The set of all these cyclic orders is called a rotation system.Embeddings with the same rotation system are considered to be equivalent and the corresponding equivalence class of embeddings is called combinatorial embedding (as opposed to the term topological embedding, which refers to the previous ...
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 ...