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A family F of graphs is said to be closed under the operation of taking minors if every minor of a graph in F also belongs to F. If F is a minor-closed family, then let S be the class of graphs that are not in F (the complement of F). According to the Robertson–Seymour theorem, there exists a finite set H of minimal elements in S.
An edge contraction is an operation that removes an edge from a graph while simultaneously merging the two vertices it used to connect. An undirected graph H is a minor of another undirected graph G if a graph isomorphic to H can be obtained from G by contracting some edges, deleting some edges, and deleting some isolated vertices.
A minor of a graph G is any graph H that is isomorphic to a graph that can be obtained from a subgraph of G by contracting some edges. If G does not have a graph H as a minor, then we say that G is H-free. Let H be a fixed graph. Intuitively, if G is a huge H-free graph, then there ought to be a "good reason" for this.
The Robertson–Seymour theorem implies that a minor-hereditary property may be characterized in terms of a finite set of forbidden minors. The term "hereditary" has been also used for graph properties that are closed with respect to taking subgraphs. [ 3 ]
This states that families of graphs closed under the graph minor operation may be characterized by a finite set of forbidden minors. As part of this work, Robertson and Seymour also proved the graph structure theorem describing the graphs in these families. [6] Additional major results in Robertson's research include the following:
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-minimal intrinsically linked graphs. However, Sachs was unable to prove that these were the only minimal linked graphs, and this was finally accomplished by Robertson, Seymour & Thomas (1995).
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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 ...