Planar Graph - Other Planarity Criteria

Other Planarity Criteria

In practice, it is difficult to use Kuratowski's criterion to quickly decide whether a given graph is planar. However, there exist fast algorithms for this problem: for a graph with n vertices, it is possible to determine in time O(n) (linear time) whether the graph may be planar or not (see planarity testing).

For a simple, connected, planar graph with v vertices and e edges, the following simple planarity criteria hold:

Theorem 1. If v ≥ 3 then e ≤ 3v − 6;

Theorem 2. If v ≥ 3 and there are no cycles of length 3, then e ≤ 2v − 4.

In this sense, planar graphs are sparse graphs, in that they have only O(v) edges, asymptotically smaller than the maximum O(v2). The graph K_3,3, for example, has 6 vertices, 9 edges, and no cycles of length 3. Therefore, by Theorem 2, it cannot be planar. Note that these theorems provide necessary conditions for planarity that are not sufficient conditions, and therefore can only be used to prove a graph is not planar, not that it is planar. If both theorem 1 and 2 fail, other methods may be used.

• Whitney's planarity criterion gives a characterization based on the existence of an algebraic dual;
• MacLane's planarity criterion gives an algebraic characterization of finite planar graphs, via their cycle spaces;
• Fraysseix–Rosenstiehl's planarity criterion gives a characterization based on the existence of a bipartition of the cotree edges of a depth-first search tree. It is central to the left-right planarity testing algorithm;
• Schnyder's theorem gives a characterization of planarity in terms of partial order dimension;
• Colin de Verdière's planarity criterion gives a characterization based on the maximum multiplicity of the second eigenvalue of certain Schrödinger operators defined by the graph.