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In graph theory, Kuratowski's theorem is a mathematical forbidden graph characterization of planar graphs, named after Kazimierz Kuratowski. It states that a finite graph is planar if and only if it does not contain a subgraph that is a subdivision of (the complete graph on five vertices) or of , (a complete bipartite graph on six vertices ...
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]
A prototypical example of this phenomenon is Kuratowski's theorem, which states that a graph is planar (can be drawn without crossings in the plane) if and only if it does not contain either of two forbidden graphs, the complete graph K 5 and the complete bipartite graph K 3,3. For Kuratowski's theorem, the notion of containment is that of ...
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. [8]
Planarity testing algorithms typically take advantage of theorems in graph theory that characterize the set of planar graphs in terms that are independent of graph drawings. These include 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 ...
Zorn's lemma is not needed for finite graphs, such as the one pictured here. Zorn's lemma , also known as the Kuratowski–Zorn lemma , is a proposition of set theory . It states that a partially ordered set containing upper bounds for every chain (that is, every totally ordered subset ) necessarily contains at least one maximal element .
Kazimierz Kuratowski (Polish pronunciation: [kaˈʑimjɛʂ kuraˈtɔfskʲi]; 2 February 1896 – 18 June 1980) was a Polish mathematician and logician.He was one of the leading representatives of the Warsaw School of Mathematics.
The clique graph of a graph is the intersection graph of its maximal cliques. Closely related concepts to complete subgraphs are subdivisions of complete graphs and complete graph minors. In particular, Kuratowski's theorem and Wagner's theorem characterize planar graphs by forbidden complete and complete bipartite subdivisions and minors ...