Editing Planar graph (section)

=== Other criteria ===

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

For a simple, connected, planar graph with {{mvar|v}} vertices and {{mvar|e}} edges and {{mvar|f}} faces, the following simple conditions hold for {{math|''v'' ≥ 3}}:

* Theorem 1. {{math|''e'' ≤ 3''v'' − 6}};
* Theorem 2. If there are no cycles of length 3, then {{math|''e'' ≤ 2''v'' − 4}}.
* Theorem 3. {{math|''f'' ≤ 2''v'' − 4}}.

In this sense, planar graphs are [[sparse graph]]s, in that they have only {{math|''O''(''v'')}} edges, asymptotically smaller than the maximum {{math|''O''(''v''{{sup|2}})}}. The graph {{math|''K''{{sub|3,3}}}}, for example, has 6 vertices, 9 edges, and no cycles of length 3.  Therefore, by Theorem 2, it cannot be planar. 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;
* [[Mac Lane's planarity criterion]] gives an algebraic characterization of finite planar graphs, via their [[cycle space]]s;
* The [[Fraysseix–Rosenstiehl 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 [[Order dimension|partial order dimension]];
* [[Colin de Verdière graph invariant|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.
* The [[Hanani–Tutte theorem]] states that a graph is planar if and only if it has a drawing in which each independent pair of edges crosses an even number of times; it can be used to characterize the planar graphs via a system of equations modulo&nbsp;2.