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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. If is a graph that contains a subgraph that is a subdivision of or ,, then is known as a Kuratowski subgraph of . [1]
A theorem similar to Kuratowski's states that a finite graph is outerplanar if and only if it does not contain a subdivision of K 4 or of K 2,3. The above is a direct corollary of the fact that a graph G is outerplanar if the graph formed from G by adding a new vertex, with edges connecting it to all the other vertices, is a planar graph.
In mathematics, the Kuratowski–Ryll-Nardzewski measurable selection theorem is a result from measure theory that gives a sufficient condition for a set-valued function to have a measurable selection function. [1] [2] [3] It is named after the Polish mathematicians Kazimierz Kuratowski and Czesław Ryll-Nardzewski. [4]
Outerplanar graphs have a forbidden graph characterization analogous to Kuratowski's theorem and Wagner's theorem for planar graphs: a graph is outerplanar if and only if it does not contain a subdivision of the complete graph K 4 or the complete bipartite graph K 2,3. [3]
Kuratowski's theorem states that a finite graph is planar if and only if it contains no subgraph homeomorphic to K 5 (complete graph on five vertices) or K 3,3 (complete bipartite graph on six vertices, three of which connect to each of the other three). In fact, a graph homeomorphic to K 5 or K 3,3 is called a Kuratowski subgraph.
Kuratowski's theorem that a graph is planar if and only if it does not contain a subgraph that is a subdivision of K 5 (the complete graph on five vertices) or K 3,3 (the utility graph, a complete bipartite graph on six vertices, three of which connect to each of the other three).
For Kuratowski's theorem, the notion of containment is that of graph homeomorphism, in which a subdivision of one graph appears as a subgraph of the other. Thus, every graph either has a planar drawing (in which case it belongs to the family of planar graphs) or it has a subdivision of at least one of these two graphs as a subgraph (in which ...
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 ...