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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.
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]
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.
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 ...
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 ...
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]