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A similar theorem states that K 4 and K 2,3 are the forbidden minors for the set of outerplanar graphs. Although the Robertson–Seymour theorem extends these results to arbitrary minor-closed graph families, it is not a complete substitute for these results, because it does not provide an explicit description of the obstruction set for any family.
Robertson has won the Fulkerson Prize three times, in 1994 for his work on the Hadwiger conjecture, in 2006 for the Robertson–Seymour theorem, and in 2009 for his proof of the strong perfect graph theorem. [11] He also won the Pólya Prize (SIAM) in 2004, the OSU Distinguished Scholar Award in 1997, and the Waterloo Alumni Achievement Medal ...
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]
The theorem is stated in the seventeenth of a series of 23 papers by Neil Robertson and Paul Seymour. Its proof is very long and involved. Its proof is very long and involved. Kawarabayashi & Mohar (2007) and Lovász (2006) are surveys accessible to nonspecialists, describing the theorem and its consequences.
Therefore, by the Robertson–Seymour theorem, the linklessly embeddable graphs have a forbidden graph characterization as the graphs that do not contain any of a finite set of minors. [ 3 ] The set of forbidden minors for the linklessly embeddable graphs was identified by Sachs (1983) : the seven graphs of the Petersen family are all minor ...
As the Robertson–Seymour theorem shows, many important families of graphs can be characterized by a finite set of forbidden minors: for instance, according to Wagner's theorem, the planar graphs are exactly the graphs that have neither the complete graph K 5 nor the complete bipartite graph K 3,3 as minors.
In the first of their famous series of papers on graph minors, Neil Robertson and Paul Seymour define a path-decomposition of a graph G to be a sequence of subsets X i of vertices of G, with two properties: For each edge of G, there exists an i such that both endpoints of the edge belong to subset X i, and
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)