Editing Kuratowski's theorem (section)

==Statement==
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 curve]]s 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 [[graph drawing|drawn]] with straight [[line segment]]s representing their edges, but by [[Fáry's theorem]] this makes no difference to their graph-theoretic characterization.

A [[subdivision (graph theory)|subdivision]] of a graph is a graph formed by subdividing its edges into [[path (graph theory)|paths]] of one or more edges. Kuratowski's theorem states that a finite graph <math>G</math> is planar if it is not possible to subdivide the edges of <math>K_5</math> or <math>K_{3,3}</math>, and then possibly add additional edges and vertices, to form a graph [[graph isomorphism|isomorphic]] to <math>G</math>. Equivalently, a finite graph is planar if and only if it does not contain a subgraph that is [[homeomorphism (graph theory)|homeomorphic]] to <math>K_5</math> or <math>K_{3,3}</math>.