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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.
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]
Category: Theorems in graph theory. 14 languages. ... Kőnig's theorem (graph theory) Kotzig's theorem; Kuratowski's theorem; M. Markov theorem; Max-flow min-cut theorem;
The autonomous development of topology from 1860 and 1930 fertilized graph theory back through the works of Jordan, Kuratowski and Whitney. Another important factor of common development of graph theory and topology came from the use of the techniques of modern algebra.
Kuratowski's research in the field of measure theory, including research with Banach and Tarski, was continued by many students. Moreover, with Alfred Tarski and Wacław Sierpiński he provided most of the theory concerning Polish spaces (that are indeed named after these mathematicians and their legacy).
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]
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 ...