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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.
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:
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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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.
Each of the Petersen family graphs forms a minimal forbidden minor for the family of YΔY-reducible graphs. [2] However, Neil Robertson provided an example of an apex graph (a linkless embeddable graph formed by adding one vertex to a planar graph) that is not YΔY-reducible, showing that the YΔY-reducible graphs form a proper subclass of the ...
Pages in category "Graph minor theory" The following 33 pages are in this category, out of 33 total. ... Robertson–Seymour theorem; S. Shallow minor; Snark (graph ...