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The Robertson–Seymour theorem has an important consequence in computational complexity, due to the proof by Robertson and Seymour that, for each fixed graph h, there is a polynomial time algorithm for testing whether a graph has h as a minor.
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 ...
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.
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 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]
By the Robertson–Seymour theorem, the graphs of branchwidth k can be characterized by a finite set of forbidden minors. The graphs of branchwidth 0 are the matchings; the minimal forbidden minors are a two-edge path graph and a triangle graph (or the two-edge cycle, if multigraphs rather than simple graphs are considered). [16]
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]
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 ...