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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.
The theory of graph minors began with Wagner's theorem that a graph is planar if and only if its minors include neither the complete graph K 5 nor the complete bipartite graph K 3,3. [1] The Robertson–Seymour theorem implies that an analogous forbidden minor characterization exists for every property of graphs that is preserved by deletions ...
Robbins theorem (graph theory) Robertson–Seymour theorem (graph theory) Robin's theorem (number theory) Robinson's joint consistency theorem (mathematical logic) Rokhlin's theorem (geometric topology) Rolle's theorem ; Rosser's theorem (number theory) Rouché's theorem (complex analysis) Rouché–Capelli theorem (Linear algebra)
A toroidal graph that cannot be embedded in a plane is said to have genus 1. The Heawood graph, the complete graph K 7 (and hence K 5 and K 6), the Petersen graph (and hence the complete bipartite graph K 3,3, since the Petersen graph contains a subdivision of it), one of the Blanuša snarks, [1] and all Möbius ladders are toroidal.
A linklessly embeddable graph is a graph that has a linkless or flat embedding; these graphs form a three-dimensional analogue of the planar graphs. [1] Complementarily, an intrinsically linked graph is a graph that does not have a linkless embedding. Flat embeddings are automatically linkless, but not vice versa. [2]
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]
Paul D. Seymour FRS (born 26 July 1950) is a British mathematician known for his work in discrete mathematics, especially graph theory.He (with others) was responsible for important progress on regular matroids and totally unimodular matrices, the four colour theorem, linkless embeddings, graph minors and structure, the perfect graph conjecture, the Hadwiger conjecture, claw-free graphs, χ ...