Ingraph theory,Kuratowski's theorem is a mathematicalforbidden graph characterization ofplanar graphs, named afterKazimierz Kuratowski. It states that a finite graph is planar if and only if it does not contain asubgraph that is asubdivision of${\displaystyle K_5}$ (thecomplete graph on fivevertices) nor of${\displaystyle K_{3,3}}$ (acomplete bipartite graph on six vertices, three of which connect to each of the other three, also known as theutility graph).

Aplanar graph is a graph whose vertices can be represented by points in theEuclidean plane, and whose edges can be represented bysimple curves in the same plane connecting the points representing their endpoints, such that no two curves intersect except at a common endpoint. Planar graphs are oftendrawn with straightline segments representing their edges, but byFáry's theorem allowing curved edges or requiring straight edges makes no difference to their graph-theoretic characterization.

Asubdivision of a graph is a graph formed by subdividing its edges intopaths of one or more edges. Kuratowski's theorem states that a finite graph${\displaystyle G}$ is planar if it is not possible to subdivide the edges of${\displaystyle K_5}$ or${\displaystyle K_{3,3}}$, and then possibly add additional edges and vertices, to form a graphisomorphic to${\displaystyle G}$. Equivalently, a finite graph is planar if and only if it does not contain a subgraph that ishomeomorphic to${\displaystyle K_5}$ or${\displaystyle K_{3,3}}$.

If${\displaystyle G}$ is a graph that contains a subgraph${\displaystyle H}$ that is a subdivision of${\displaystyle K_5}$ or${\displaystyle K_{3,3}}$, then${\displaystyle H}$ is known as aKuratowski subgraph of${\displaystyle G}$.^[1] 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${\displaystyle K_5}$ and${\displaystyle K_{3,3}}$ are nonplanar, as may be shown either by acase analysis or an argument involvingEuler's formula. Additionally, subdividing a graph cannot turn a nonplanar graph into a planar graph: if a subdivision of a graph${\displaystyle G}$ has a planar drawing, the paths of the subdivision form curves that may be used to represent the edges of${\displaystyle G}$ itself. Therefore, a graph that contains a Kuratowski subgraph cannot be planar. The more difficult direction in proving Kuratowski's theorem is to show that, if a graph is nonplanar, it must contain a Kuratowski subgraph.

A Kuratowski subgraph of a nonplanar graph can be found inlinear time, as measured by the size of the input graph.^[2] This allows the correctness of aplanarity testing algorithm to be verified for nonplanar inputs, as it is straightforward to test whether a given subgraph is or is not a Kuratowski subgraph.^[3]Usually, non-planar graphs contain a large number of Kuratowski-subgraphs. The extraction of these subgraphs is needed, e.g., inbranch and cut algorithms for crossing minimization. It is possible to extract a large number of Kuratowski subgraphs in time dependent on their total size.^[4]

Kazimierz Kuratowski published his theorem in 1930.^[5] The theorem was independently proved byOrrin Frink andPaul Smith, also in 1930,^[6] but their proof was never published. The special case ofcubic planar graphs (for which the only minimal forbidden subgraph is${\displaystyle K_{3,3}}$) was also independently proved byKarl Menger in 1930.^[7]Since then, several new proofs of the theorem have been discovered.^[8]

In theSoviet Union, Kuratowski's theorem was known as either thePontryagin–Kuratowski theorem or theKuratowski–Pontryagin theorem,^[9]as the theorem was reportedly proved independently byLev Pontryagin around 1927.^[10]However, as Pontryagin never published his proof, this usage has not spread to other places.^[11]

A closely related result,Wagner's theorem, characterizes the planar graphs by theirminors in terms of the same two forbidden graphs${\displaystyle K_5}$ and${\displaystyle K_{3,3}}$. Every Kuratowski subgraph is a special case of a minor of the same type, and while the reverse is not true, it is not difficult to find a Kuratowski subgraph (of one type or the other) from one of these two forbidden minors; therefore, these two theorems are equivalent.^[12]

An extension is theRobertson–Seymour theorem, stating that every class of graphs closed under taking minors (as the planar graphs are) can be characterized in an analogous way by a finite set of forbidden minors.

• Kelmans–Seymour conjecture, that 5-connected nonplanar graphs contain a subdivision of${\displaystyle K_5}$

• Statements about planar graphs
• Theorems in graph theory

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