Kuratowski's theorem
It states that a finite graph is planar if and only if it does not contain a subgraph that is a subdivision of
A planar graph is a graph whose vertices can be represented by points in the Euclidean plane, and whose edges can be represented by simple 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 often drawn with straight line segments representing their edges, but by Fáry's theorem this makes no difference to their graph-theoretic characterization.
Kuratowski's theorem states that a finite graph
, and then possibly add additional edges and vertices, to form a graph isomorphic to
Equivalently, a finite graph is planar if and only if it does not contain a subgraph that is homeomorphic to
are nonplanar, as may be shown either by a case analysis or an argument involving Euler's formula.
has a planar drawing, the paths of the subdivision form curves that may be used to represent the edges of
[2] This allows the correctness of a planarity 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., in branch and cut algorithms for crossing minimization.
It is possible to extract a large number of Kuratowski subgraphs in time dependent on their total size.
[5] The theorem was independently proved by Orrin Frink and Paul Smith, also in 1930,[6] but their proof was never published.
The special case of cubic planar graphs (for which the only minimal forbidden subgraph is
[10] However, as Pontryagin never published his proof, this usage has not spread to other places.
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.
