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 Connections
Has Statement Form
Entity-backed identity for the statement-form enum value this theorem already carries, resolved to a mathematics concept by an explicit value-to-entity map (phase 3 bucket conversion, docs\design_entity_backed_browse_buckets_20260928.md). The statement-form fact itself stays on the theorem unchanged.
In Branch
Sources
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.
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