Planar Graphs

A planar graph is a graph that can be drawn in a plane without any of its edges crossing each other. Drawing a graph in the plane without edge crossings is called embedding the graph in the plane. Planar graphs are important in circuit design, map coloring, and network layout problems.

Planar Graph

A graph G is called planar if it can be drawn in a plane without any edges crossing. The same graph may have multiple drawings − some with crossings and some without. If at least one crossing-free drawing exists, the graph is planar.

Example

The complete graph K₄ is planar because it can be embedded in the plane with no edge crossings. The edge from c to b is drawn as a curve around the outside to avoid crossing the diagonal edge.

Non-Planar Graph

A graph is non-planar if it cannot be drawn in a plane without graph edges crossing, no matter how the vertices and edges are arranged.

Example

The complete graph K₅ is non-planar. No matter how you arrange the 5 vertices, at least one pair of edges must cross. The dashed lines show the inner edges that inevitably produce crossings.

Key Facts About Planar Graphs

By Kuratowski's theorem, a graph is non-planar if and only if it contains a subgraph that is homeomorphic to K₅ (complete graph on 5 vertices) or K_3,3 (complete bipartite graph on 3+3 vertices). Some useful planarity facts −

Conclusion

A graph is planar if it can be embedded in a plane with no edge crossings, and non-planar otherwise. The smallest non-planar graphs are K₅ and K_3,3, and Kuratowski's theorem uses these as the basis for testing planarity of any graph.