# Planar Graph **A [[graph]] which can be depicted on a plane such that none of the edges are intersecting.** ![[PlanarNon-PlanarGraph.svg|600]] *The [[Complete Graph|complete graphs]] $K_{4}$ and $K_{5}$ which are planar and non-planar, respectively* > [!Example] Planar Graph > A *planar graph* is a graph that can be *embedded* in the *plane*. > > Alternatively, it is a graph that can be depicted on a plane such that its edges *only intersect* at their *endpoints*. A [[Connected Graph|connected]] planar graph drawn such that none of its edges intersect has an [[Euler characteristic]] of $2$ which is also shared with all *convex polyhedra*. As such, the [[Schlegel Diagram|Schlegel diagrams]] of convex polyhedra are connected planar graphs. ## Alternative criteria > [!Example] Kuratowski's Theorem > *Kuratowski's theorem* states that a finite graph is planar *if and only if* it does not contain a subgraph that is a *subdivision*[^1] of the complete graph $K_5$ and the complete [[bipartite graph]] $K_{3,3}$. > [!Example] Wagner's Theorem > *Wagner's theorem* states that a finite graph is planar *if and only if* it has neither $K_5$ nor $K_{3,3}$ as a *minor*[^2]. [^1]: A *subdivision* of a graph is a graph formed from the insertion of zero or more vertices into edges, thus dividing an edge into several. [^2]: A *minor* of a graph is a graph formed from the deletion of edges, and subsequently isolated vertices, and the [[Graph Condensation#Edge Contraction|contraction of edges]].