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to a minor of another. The truth of this conjecture implies that any family of graphs closed under the operation of taking minors (as planar graphs are) can automatically be characterized by
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with three vertices on each side of its bipartition. That is, these two graphs are the only minor-minimal non-planar graphs. It is closely related to, but should be distinguished from,
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of up to three vertices and then possibly remove edges from those cliques. This characterization was used by Wagner to show that the case
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on graph theory, and in June 2000, following Wagner's death, the
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finally published a proof of Wagner's conjecture in 2004 and it is now known as the
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Contemporary
Methods in Graph Theory: In honour of Prof. Dr. Klaus Wagner
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analogously to Wagner's theorem characterizing the planar graphs.
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Another result of his, also known as Wagner's theorem, is that a
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minor. This implies a characterization of the graphs with no
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as exactly those graphs that do not have as a minor either a
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278:four-connected graph
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227:characterizes the
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386:: 570–590,
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356:Recognition
314:= 5 of the
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115:Mathematics
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157:Karl Dörge
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149:topology
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