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With this notation, Kuratowski's theorem can be expressed succinctly: a graph is planar if and only if it does not have a Kuratowski subgraph. The two graphs K 5 {\displaystyle K_{5}} and K 3 , 3 {\displaystyle K_{3,3}} are nonplanar, as may be shown either by a case analysis or an argument involving Euler's formula .
The Polish mathematician Kazimierz Kuratowski provided a characterization of planar graphs in terms of forbidden graphs, now known as Kuratowski's theorem: A finite graph is planar if and only if it does not contain a subgraph that is a subdivision of the complete graph K 5 or the complete bipartite graph K 3,3 (utility graph).
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 ...
For example, Kuratowski's Theorem states: A graph is planar if it contains as a subdivision neither the complete bipartite graph K 3,3 nor the complete graph K 5. Another problem in subdivision containment is the Kelmans–Seymour conjecture: Every 5-vertex-connected graph that is not planar contains a subdivision of the 5-vertex complete graph ...
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 ...
In graph theory, two graphs and ′ are homeomorphic if there is a graph isomorphism from some subdivision of to some subdivision of ′.If the edges of a graph are thought of as lines drawn from one vertex to another (as they are usually depicted in diagrams), then two graphs are homeomorphic to each other in the graph-theoretic sense precisely if their diagrams are homeomorphic in the ...
Multiple proofs of this impossibility are known, and form part of the proof of Kuratowski's theorem characterizing planar graphs by two forbidden subgraphs, one of which is ,. The question of minimizing the number of crossings in drawings of complete bipartite graphs is known as Turán's brick factory problem , and for K 3 , 3 {\displaystyle K ...
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 .