Editing Planar graph (section)

== Planarity criteria ==
=== Kuratowski's and Wagner's theorems ===
{{tesseract_graph_nonplanar_visual_proof.svg}}
The [[Poland|Polish]] mathematician [[Kazimierz Kuratowski]] provided a characterization of planar graphs in terms of [[Forbidden graph characterization|forbidden graphs]], now known as [[Kuratowski's theorem]]:

:A [[finite graph]] is planar [[if and only if]] it does not contain a [[Glossary of graph theory#subgraph|subgraph]] that is a [[subdivision (graph theory)|subdivision]] of the [[complete graph]] {{math|''K''{{sub|5}}}} or the [[complete bipartite graph]] {{math|''K''{{sub|3,3}}}} ([[utility graph]]).

A [[subdivision (graph theory)|subdivision]] of a graph results from inserting vertices  into edges (for example, changing an edge {{nowrap|• —— • to • — • — • )}} zero or more times.
[[Image:Nonplanar no subgraph K 3 3.svg|thumb|An example of a graph with no {{math|''K''{{sub|5}}}} or {{math|''K''{{sub|3,3}}}} subgraph. However, it contains a subdivision of {{math|''K''{{sub|3,3}}}} and is therefore non-planar.]]

Instead of considering subdivisions, [[Wagner's theorem]] deals with [[minor (graph theory)|minors]]:

:A finite graph is planar if and only if it does not have {{math|''K''{{sub|5}}}} or {{math|''K''{{sub|3,3}}}} as a [[minor (graph theory)|minor]].

A [[minor (graph theory)|minor]] of a graph results from taking a subgraph and repeatedly contracting an edge into a vertex, with each neighbor of the original end-vertices becoming a neighbor of the new vertex.

[[File:Kuratowski.gif|thumb|484px|An animation showing that the [[Petersen graph]] contains a minor isomorphic to the {{math|''K''{{sub|3,3}}}} graph, and is therefore non-planar]]

[[Klaus Wagner (mathematician)|Klaus Wagner]] asked more generally whether any minor-closed class of graphs is determined by a finite set of "[[forbidden minor]]s". This is now the [[Robertson–Seymour theorem]], proved in a long series of papers. In the language of this theorem, {{math|''K''{{sub|5}}}} and {{math|''K''{{sub|3,3}}}} are the forbidden minors for the class of finite planar graphs.

=== 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.