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Some examples of finite obstruction sets were already known for specific classes of graphs before the Robertson–Seymour theorem was proved. For example, the obstruction for the set of all forests is the loop graph (or, if one restricts to simple graphs, the cycle with three vertices). This means that a graph is a forest if and only if none of ...
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.
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]
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 ...
One example is the snark theorem, that every cubic graph requiring four colors in any edge coloring has the Petersen graph as a minor, conjectured by W. T. Tutte and announced to be proved in 2001 by Robertson, Sanders, Seymour, and Thomas. [13]
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 ...
The proof of the strong perfect graph theorem by Chudnovsky et al. follows an outline conjectured in 2001 by Conforti, Cornuéjols, Robertson, Seymour, and Thomas, according to which every Berge graph either forms one of five types of basic building block (special classes of perfect graphs) or it has one of four different types of structural ...
A graph with three vertices and three edges. A graph (sometimes called an undirected graph to distinguish it from a directed graph, or a simple graph to distinguish it from a multigraph) [4] [5] is a pair G = (V, E), where V is a set whose elements are called vertices (singular: vertex), and E is a set of unordered pairs {,} of vertices, whose elements are called edges (sometimes links or lines).