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The existence of forbidden minor characterizations for all minor-closed graph families is an equivalent way of stating the Robertson–Seymour theorem. For, suppose that every minor-closed family F has a finite set H of minimal forbidden minors, and let S be any infinite set of graphs. Define F from S as the family of graphs that do not have a ...
A graph H is called a topological minor of a graph G if a subdivision of H is isomorphic to a subgraph of G. [21] Every topological minor is also a minor. The converse however is not true in general (for instance the complete graph K 5 in the Petersen graph is a minor but not a topological one), but holds for graph with maximum degree not ...
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:
Graph minor Wagner's theorem: Outerplanar graphs: K 4 and K 2,3: Graph minor Diestel (2000), [1] p. 107: Outer 1-planar graphs: Six forbidden minors Graph minor Auer et al. (2013) [2] Graphs of fixed genus: A finite obstruction set Graph minor Diestel (2000), [1] p. 275: Apex graphs: A finite obstruction set Graph minor [3] Linklessly ...
In mathematics, a simple subcubic graph (SSCG) is a finite simple graph in which each vertex has a degree of at most three. Suppose we have a sequence of simple subcubic graphs G 1 , G 2 , ... such that each graph G i has at most i + k vertices (for some integer k ) and for no i < j is G i homeomorphically embeddable into (i.e. is a graph minor ...
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An animation showing that the Petersen graph contains a minor isomorphic to the K 3,3 graph, and is therefore non-planar. Klaus Wagner asked more generally whether any minor-closed class of graphs is determined by a finite set of "forbidden minors". This is now the Robertson–Seymour theorem, proved in a long series of papers.
The algorithm is for testing membership of a graph G in a minor-closed family F, not for testing whether a family F is minor-closed. The algorithm is: for each obstruction H for F, test whether H is a minor of G. That part is concrete.