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A subdivision of K 3,3 in the generalized Petersen graph G(9,2), showing that the graph is nonplanar.. In graph theory, Kuratowski's theorem is a mathematical forbidden graph characterization of planar graphs, named after Kazimierz Kuratowski.
When at most three regions meet at a point, the result is a planar graph, but when four or more regions meet at a point, the result can be nonplanar (for example, if one thinks of a circle divided into sectors, with the sectors being the regions, then the corresponding map graph is the complete graph as all the sectors have a common boundary ...
Proof without words that a hypercube graph is non-planar using Kuratowski's or Wagner's theorems and finding either K 5 (top) or K 3,3 (bottom) subgraphs. Wagner published both theorems in 1937, [1] subsequent to the 1930 publication of Kuratowski's theorem, [2] according to which a graph is planar if and only if it does not contain as a subgraph a subdivision of one of the same two forbidden ...
Conversely, every nonplanar graph contains either K 3,3 or the complete graph K 5 as a minor; this is Wagner's theorem. [9] Every complete bipartite graph. K n,n is a Moore graph and a (n,4)-cage. [10]
More generally, a forbidden graph characterization is a method of specifying a family of graph, or hypergraph, structures, by specifying substructures that are forbidden to exist within any graph in the family.
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A related result, Wagner's theorem, states that every 4-vertex-connected nonplanar graph contains a copy of K 5 as a graph minor. One way of restating this result is that, in these graphs, it is always possible to perform a sequence of edge contraction operations so that the resulting graph contains a K 5 subdivision. The Kelmans–Seymour ...