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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.
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.
In the language of the later papers in Robertson and Seymour's graph minor series, a path-decomposition is a tree decomposition (X,T) in which the underlying tree T of the decomposition is a path graph.
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.
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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:
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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).