A finite graph is planar, meaning it can be drawn in the plane without edge crossings, if and only if it contains no subgraph that is a subdivision of the complete graph on five vertices or the complete bipartite graph on three plus three vertices. Proved by Kazimierz Kuratowski, it is the classical characterization of planarity in graph theory.
Facts
StatementA finite graph is planar if and only if it does not contain a subgraph that is a subdivision of K5 or of K3,3; equivalently, a graph is planar if and only if it has no Kuratowski subgraph. 2 Classification
Statement FormCharacterization Theorem 1 Kuratowski's Theorem
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Source MacTutor History of Mathematics, University of St Andrews: Kazimierz Kuratowski
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Source Kuratowski's Theorem (Wikipedia)
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1. Wikipedia: Kuratowski's theorem
WikipediaLead section, statement-form referenceQuote, Lead section, statement-form reference
It states that a finite 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) nor of K 3, 3 } (a complete bipartite graph on six vertices, three of which connect to each of the other three, also known as the utility graph).
2. Kuratowski's Theorem (Wikipedia)
Wikimedia FoundationKuratowski subgraphs section
Kuratowski's theorem can be expressed succinctly: a graph is planar if and only if it does not have a Kuratowski subgraph.
History section
Kazimierz Kuratowski published his theorem in 1930.
Proved By: Kazimierz Kuratowski, Lead
Kazimierz Kuratowski published his theorem in 1930.
In Group: Graph Theory, Lead section
In graph theory, Kuratowski's theorem is a mathematical forbidden
View the Source MacTutor History of Mathematics, University of St Andrews: Kazimierz Kuratowski
MacTutor History of Mathematics, University of St AndrewsPrincipal Result Of: Kazimierz Kuratowski, Biography textQuote, Principal Result Of: Kazimierz Kuratowski, Biography text
He is best known for his theorem giving a necessary and sufficient condition for a graph to be planar.
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