Editing Planar graph

{{Redirect|Triangular graph|line graphs of complete graphs|Line graph#Strongly regular and perfect line graphs|triangulated graphs|Chordal graph|data graphs plotted across three variables|Ternary plot}}
{{short description|Graph that can be embedded in the plane}}
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{|class="wikitable" align="right" style="margin-left: 1em;"
!colspan="2"|Example graphs
|-
! Planar
! Nonplanar
|-
| align="center" | [[Image:Butterfly graph.svg|100px]] <br> [[Butterfly graph]]
| align="center" | [[Image:Complete graph K5.svg|100px]] <br>[[Complete graph]] ''K''<sub>5</sub>
|-
| align="center" | [[File:CGK4PLN.svg|100x100px]] <br> [[Complete graph]]<br> ''K''<sub>4</sub> 
| align="center" | [[Image:Biclique K 3 3.svg|100px]] <br>[[Utility graph]] ''K''<sub>3,3</sub>
|}

In [[graph theory]], a '''planar graph''' is a [[graph (discrete mathematics)|graph]] that can be [[graph embedding|embedded]] in the [[plane (geometry)|plane]], i.e., it can be drawn on the plane in such a way that its edges intersect only at their endpoints.  In other words, it can be drawn in such a way that no edges cross each other.<ref>{{cite book|last=Trudeau|first=Richard J.|title=Introduction to Graph Theory|year=1993|publisher=Dover Pub.|location=New York|isbn=978-0-486-67870-2|pages=64|url=http://store.doverpublications.com/0486678709.html|edition=Corrected, enlarged republication.|access-date=8 August 2012|quote=Thus a planar graph, when drawn on a flat surface, either has no edge-crossings or can be redrawn without them.}}</ref><ref>{{cite book |last1=Barthelemy |first1=M. |chapter=1.5 Planar Graphs |chapter-url={{GBurl|9-hEDwAAQBAJ|p=6}}  |title=Morphogenesis of Spatial Networks |date=2017 |isbn=978-3-319-20565-6 |publisher=Springer |page=6}}</ref>   Such a drawing is called a '''plane graph''', or a '''planar embedding''' of the graph. A plane graph can be defined as a planar graph with a mapping from every node to a point on a plane, and from every edge to a [[plane curve]] on that plane, such that the extreme points of each curve are the points mapped from its end nodes, and all curves are disjoint except on their extreme points.

Every graph that can be drawn on a plane can be drawn on the [[sphere]] as well, and vice versa, by means of [[stereographic projection]].

Plane graphs can be encoded by [[combinatorial map]]s or [[rotation system]]s.

An [[equivalence class]] of [[topologically equivalent]] drawings on the sphere, usually with additional assumptions such as the absence of [[bridge (graph theory)|isthmus]]es, is called a '''planar map'''. Although a plane graph has an '''external''' or '''unbounded''' [[face (graph theory)|face]], none of the faces of a planar map has a particular status.

Planar graphs generalize to graphs drawable on a surface of a given [[genus (mathematics)|genus]]. In this terminology, planar graphs have [[graph genus|genus]]&nbsp;0, since the plane (and the sphere) are surfaces of genus&nbsp;0. See "[[graph embedding]]" for other related topics.

== 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.

== Properties ==
=== Euler's formula ===
<!--Linked to from [[Crossing number inequality#Proof]]-->
{{main|Euler characteristic#Plane graphs}}
'''Euler's formula''' states that if a finite, [[Connectivity (graph theory)|connected]], planar graph is drawn in the plane without any edge intersections, and {{mvar|v}} is the number of vertices, {{mvar|e}} is the number of edges and {{mvar|f}} is the number of faces (regions bounded by edges, including the outer, infinitely large region), then

:<math>v-e+f=2.</math>

As an illustration, in the [[butterfly graph]] given above, {{math|1=''v'' = 5}}, {{math|1=''e'' = 6}} and {{math|1=''f'' = 3}}. 
In general, if the property holds for all planar graphs of {{mvar|f}} faces, any change to the graph that creates an additional face while keeping the graph planar would keep {{math|''v'' − ''e'' + ''f''}} an invariant. Since the property holds for all graphs with {{math|1=''f'' = 2}}, by [[mathematical induction]] it holds for all cases. Euler's formula can also be proved as follows: if the graph isn't a [[tree (graph theory)|tree]], then remove an edge which completes a [[cycle (graph theory)|cycle]]. This lowers both {{mvar|e}} and {{mvar|f}} by one, leaving {{math|''v'' − ''e'' + ''f''}} constant. Repeat until the remaining graph is a tree; trees have {{math|1=''v'' = ''e'' + 1}} and {{math|1=''f'' = 1}}, yielding {{math|1=''v'' − ''e'' + ''f'' = 2}}, i. e., the [[Euler characteristic]] is 2.

In a finite, [[Connectivity (graph theory)|connected]], ''[[simple graph|simple]]'', planar graph, any face (except possibly the outer one) is bounded by at least three edges and every edge touches at most two faces, so {{math|1=3''f'' ≤ 2''e''}}; using Euler's formula, one can then show that these graphs are ''sparse'' in the sense that if {{math|''v'' ≥ 3}}:

:<math>e\leq 3v-6.</math>

[[File:Dodecahedron schlegel.svg|thumb|A [[Schlegel diagram]] of a regular [[dodecahedron]], forming a planar graph from a convex polyhedron.]]

Euler's formula is also valid for [[convex polyhedron|convex polyhedra]]. This is no coincidence: every convex polyhedron can be turned into a connected, simple, planar graph by using the [[Schlegel diagram]] of the polyhedron, a [[perspective projection]] of the polyhedron onto a plane with the center of perspective chosen near the center of one of the polyhedron's faces. Not every planar graph corresponds to a convex polyhedron in this way: the trees do not, for example. [[Steinitz's theorem]] says that the [[polyhedral graph]]s formed from convex polyhedra are precisely the finite [[Connectivity (graph theory)|3-connected]] simple planar graphs. More generally, Euler's formula applies to any polyhedron whose faces are [[simple polygon]]s that form a surface [[homeomorphism|topologically equivalent]] to a sphere, regardless of its convexity.

=== Average degree ===

Connected planar graphs with more than one edge obey the inequality {{math|2''e'' ≥ 3''f''}}, because each face has at least three face-edge incidences and each edge contributes exactly two incidences. It follows via algebraic transformations of this inequality with Euler's formula {{math|1=''v'' − ''e'' + ''f'' = 2}} that for finite planar graphs the average degree is strictly less than 6. Graphs with higher average degree cannot be planar.

=== Coin graphs ===
{{main|Circle packing theorem}}
[[File:Circle packing theorem K5 minus edge example.svg|thumb|Example of the circle packing theorem on {{math|''K''{{sup| −}}{{sub|5}}}}, the complete graph on five vertices, minus one edge.]]

We say that two circles drawn in a plane ''kiss'' (or ''[[Osculating circle|osculate]]'') whenever they intersect in exactly one point.  A "coin graph" is a graph formed by a set of circles, no two of which have overlapping interiors, by making a vertex for each circle and an edge for each pair of circles that kiss. The [[circle packing theorem]], first proved by [[Paul Koebe]] in 1936,  states that a graph is planar if and only if it is a coin graph.

This result provides an easy proof of [[Fáry's theorem]], that every simple planar graph can be embedded in the plane in such a way that its edges are straight [[line segment]]s that do not cross each other. If one places each vertex of the graph at the center of the corresponding circle in a coin graph representation, then the line segments between centers of kissing circles do not cross any of the other edges.

=== Planar graph density ===
The [[meshedness coefficient]] or density {{mvar|D}} of a planar graph, or network, is the ratio of the number {{math|''f'' − 1}} of bounded faces (the same as the [[circuit rank]] of the graph, by [[Mac Lane's planarity criterion]]) by its maximal possible values {{math|2''v'' − 5}} for a graph with {{mvar|v}} vertices:
:<math>D = \frac{f-1}{2v-5}</math>
The density obeys {{math|0 ≤ ''D'' ≤ 1}}, with {{math|1=''D'' = 0}} for 
a completely sparse planar graph (a tree), and {{math|1=''D'' = 1}} for a completely dense (maximal) planar graph.<ref>{{citation
 | last1 = Buhl | first1 = J.
 | last2 = Gautrais | first2 = J.
 | last3 = Sole | first3 = R.V.
 | last4 = Kuntz | first4 = P.
 | last5 = Valverde | first5 = S.
 | last6 = Deneubourg | first6 = J.L.
 | last7 = Theraulaz | first7 = G.
 | doi = 10.1140/epjb/e2004-00364-9
 | issue = 1
 | journal = European Physical Journal B
 | pages = 123–129
 | title = Efficiency and robustness in ant networks of galleries
 | volume = 42
 | year = 2004| bibcode = 2004EPJB...42..123B| s2cid = 14975826
 }}.</ref>

===Dual graph===
[[Image:dual graphs.svg|thumb|100px|A planar graph and its [[Dual graph|dual]]]]

Given an embedding {{mvar|G}} of a (not necessarily simple) connected graph in the plane without edge intersections, we construct the ''[[dual graph]]'' {{mvar|G*}} as follows: we choose one vertex in each face of {{mvar|G}} (including the outer face) and for each edge {{mvar|e}} in {{mvar|G}} we introduce a new edge in {{mvar|G*}} connecting the two vertices in {{mvar|G*}} corresponding to the two faces in {{mvar|G}} that meet at {{mvar|e}}. Furthermore, this edge is drawn so that it crosses {{mvar|e}} exactly once and that no other edge of {{mvar|G}} or {{mvar|G*}} is intersected. Then {{mvar|G*}} is again the embedding of a (not necessarily simple) planar graph; it has as many edges as {{mvar|G}}, as many vertices as {{mvar|G}} has faces and as many faces as {{mvar|G}} has vertices. The term "dual" is justified by the fact that {{math|1=''G''** = ''G''}}; here the equality is the equivalence of embeddings on the [[sphere]]. If {{mvar|G}} is the planar graph corresponding to a convex polyhedron, then {{mvar|G*}} is the planar graph corresponding to the dual polyhedron.

Duals are useful because many properties of the dual graph are related in simple ways to properties of the original graph, enabling results to be proven about graphs by examining their dual graphs.

While the dual constructed for a particular embedding is unique (up to [[isomorphism]]), graphs may have different (i.e. non-isomorphic) duals, obtained from different (i.e. non-[[homeomorphic]]) embeddings.

==Families of planar graphs==

===Maximal planar graphs===
[[File:Goldner-Harary graph.svg|thumb|240px|The [[Goldner–Harary graph]] is maximal planar. All its faces are bounded by three edges.]]

A simple graph is called '''maximal planar''' if it is planar but adding any edge (on the given vertex set) would destroy that property. All faces (including the outer one) are then bounded by three edges, explaining the alternative term '''plane triangulation''' (which technically means a plane drawing of the graph). The alternative names "triangular graph"<ref>{{citation|first=W.|last=Schnyder|title=Planar graphs and poset dimension|journal=[[Order (journal)|Order]]|volume=5|year=1989|issue=4|pages=323–343|doi=10.1007/BF00353652|mr=1010382|s2cid=122785359}}.</ref> or "triangulated graph"<ref>{{citation|journal=Algorithmica|volume=3|issue=1–4|year=1988|pages=247–278|doi= 10.1007/BF01762117|title=A linear algorithm to find a rectangular dual of a planar triangulated graph|first1=Jayaram|last1=Bhasker|first2=Sartaj|last2=Sahni|s2cid=2709057}}.</ref> have also been used, but are ambiguous, as they more commonly refer to the [[line graph]] of a [[complete graph]] and to the [[chordal graph]]s respectively. Every maximal planar graph on more than 3 vertices is at least 3-connected.<ref>{{citation
 | last1 = Hakimi | first1 = S. L.
 | last2 = Schmeichel | first2 = E. F.
 | doi = 10.1002/jgt.3190020404
 | issue = 4
 | journal = Journal of Graph Theory
 | mr = 512801
 | pages = 307–314
 | title = On the connectivity of maximal planar graphs
 | volume = 2
 | year = 1978}}; Hakimi and Schmeichel credit the 3-connectivity of maximal planar graphs to a theorem of [[Hassler Whitney]].</ref>

If a maximal planar graph has {{mvar|v}} vertices with {{math|''v'' > 2}}, then it has precisely {{math|3''v'' − 6}} edges and {{math|2''v'' − 4}} faces.

[[Apollonian network]]s are the maximal planar graphs formed by repeatedly splitting triangular faces into triples of smaller triangles.  Equivalently, they are the planar [[k-tree|3-trees]].

[[Strangulated graph]]s are the graphs in which every [[peripheral cycle]] is a triangle. In a maximal planar graph (or more generally a polyhedral graph) the peripheral cycles are the faces, so maximal planar graphs are strangulated. The strangulated graphs include also the [[chordal graph]]s, and are exactly the graphs that can be formed by [[clique-sum]]s (without deleting edges) of [[complete graph]]s and maximal planar graphs.<ref>{{citation
 | last1 = Seymour | first1 = P. D.
 | last2 = Weaver | first2 = R. W.
 | doi = 10.1002/jgt.3190080206
 | issue = 2
 | journal = Journal of Graph Theory
 | mr = 742878
 | pages = 241–251
 | title = A generalization of chordal graphs
 | volume = 8
 | year = 1984}}.</ref>

===Outerplanar graphs===
[[Outerplanar graph]]s are graphs with an embedding in the plane such that all vertices belong to the unbounded face of the embedding. Every outerplanar graph is planar, but the converse is not true: {{math|''K''<sub>4</sub>}} is planar but not outerplanar. A theorem similar to Kuratowski's states that a finite graph is outerplanar if and only if it does not contain a subdivision of {{math|''K''<sub>4</sub>}} or of {{math|''K''<sub>2,3</sub>}}. The above is a direct corollary of the fact that a graph {{mvar|G}} is outerplanar if the graph formed from {{mvar|G}} by adding a new vertex, with edges connecting it to all the other vertices, is a planar graph.<ref>{{citation
 | last = Felsner | first = Stefan
 | contribution = 1.4 Outerplanar Graphs and Convex Geometric Graphs
 | doi = 10.1007/978-3-322-80303-0_1
 | isbn = 3-528-06972-4
 | mr = 2061507
 | pages = 6–7
 | publisher = Friedr. Vieweg & Sohn, Wiesbaden
 | series = Advanced Lectures in Mathematics
 | title = Geometric graphs and arrangements
 | year = 2004}}</ref>

A 1-outerplanar embedding of a graph is the same as an outerplanar embedding.  For {{math|''k'' > 1}} a planar embedding is {{mvar|k}}-outerplanar if removing the vertices on the outer face results in a {{math|(''k'' − 1)}}-outerplanar embedding.  A graph is {{mvar|k}}-outerplanar if it has a {{mvar|k}}-outerplanar embedding.

===Halin graphs===
A [[Halin graph]] is a graph formed from an undirected plane tree (with no degree-two nodes) by connecting its leaves into a cycle, in the order given by the plane embedding of the tree. Equivalently, it is a [[polyhedral graph]] in which one face is adjacent to all the others. Every Halin graph is planar. Like outerplanar graphs, Halin graphs have low [[treewidth]], making many algorithmic problems on them more easily solved than in unrestricted planar graphs.<ref>{{citation
 | last1 = Sysło | first1 = Maciej M.
 | last2 = Proskurowski | first2 = Andrzej
 | contribution = On Halin graphs
 | doi = 10.1007/BFb0071635
 | pages = 248–256
 | publisher = Springer-Verlag
 | series = Lecture Notes in Mathematics
 | title = Graph Theory: Proceedings of a Conference held in Lagów, Poland, February 10–13, 1981
 | volume = 1018
 | isbn = 978-3-540-12687-4
 | year = 1983}}.</ref>

===Upward planar graphs===
An [[Upward planar drawing|upward planar graph]] is a [[directed acyclic graph]] that can be drawn in the plane with its edges as non-crossing curves that are consistently oriented in an upward direction. Not every planar directed acyclic graph is upward planar, and it is [[NP-complete]] to test whether a given graph is upward planar.

===Convex planar graphs===
A planar graph is said to be '''convex''' if all of its faces (including the outer face) are [[convex polygon]]s. Not all planar graphs have a convex embedding (e.g. the complete bipartite graph {{math|K{{sub|2,4}}}}). A sufficient condition that a graph can be drawn convexly is that it is a [[Subdivision (graph theory)|subdivision]] of a [[k-vertex-connected graph|3-vertex-connected]] planar graph. [[Tutte embedding|Tutte's spring theorem]] even states that for simple 3-vertex-connected planar graphs the position of the inner vertices can be chosen to be the average of its neighbors.

===Word-representable planar graphs===
[[Word-representable graph|Word-representable]] planar graphs include triangle-free planar graphs and, more generally, 3-colourable planar graphs,<ref name="HK|last=P16">{{cite journal |first1=M. |last1=Halldórsson |first2=S. |last2=Kitaev |first3=A. |last3=Pyatkin. |title=Semi-transitive orientations and word-representable graphs |journal=Discr. Appl. Math.  |volume=201  |issue= |pages=164–171  |doi= 10.1016/j.dam.2015.07.033|date=2016 |s2cid=26796091 |doi-access=free |url=https://strathprints.strath.ac.uk/54148/2/Halldorsson_etal_DAM_2015_Semi_transitive_orientations_and_word_representable.pdf }}</ref> as well as certain face subdivisions of triangular grid graphs,<ref name="CKS2016">{{cite journal |arxiv=1503.08002 |first1=T. Z. Q. |last1=Chen |first2=S. |last2=Kitaev |first3=B. Y. |last3=Sun |title=Word-representability of face subdivisions of triangular grid graphs |journal=Graphs and Combin |volume=32 |issue=5 |pages=1749–61 |date=2016 |doi= 10.1007/s00373-016-1693-z|s2cid=43817300 |url=}}</ref> and certain triangulations of grid-covered cylinder graphs.<ref name="CKS2016-2">{{cite journal |arxiv=1507.06749 |first1=T. Z. Q. |last1=Chen |first2=S. |last2=Kitaev |first3=B. Y. |last3=Sun |title=Word-representability of triangulations of grid-covered cylinder graphs |journal=Discr. Appl. Math. |volume=213 |issue= |pages=60–70 |date=2016 |doi= 10.1016/j.dam.2016.05.025|s2cid=26987743 |url=}}</ref>

==Theorems==
=== Enumeration of planar graphs ===
The [[Asymptotic analysis|asymptotic]] for the number of (labeled) planar graphs on <math>n</math> vertices is <math>g\cdot n^{-7/2}\cdot \gamma^n\cdot n!</math>, where <math>\gamma\approx 27.22687</math> and <math>g\approx 0.43\times 10^{-5}</math>.<ref>{{cite journal | last1 = Giménez | first1 = Omer | last2 = Noy | first2 = Marc | year = 2009 | title = Asymptotic enumeration and limit laws of planar graphs | journal = Journal of the American Mathematical Society | volume = 22 | issue = 2| pages = 309–329 | doi=10.1090/s0894-0347-08-00624-3| arxiv = math/0501269 | bibcode = 2009JAMS...22..309G | s2cid = 3353537 }}</ref>

Almost all planar graphs have an exponential number of automorphisms.<ref>{{cite journal | last1 = McDiarmid | first1 = Colin | last2 = Steger | first2 = Angelika | author2-link = Angelika Steger | last3 = Welsh | first3 = Dominic J.A. | year = 2005| title = Random planar graphs | journal = Journal of Combinatorial Theory, Series B | volume = 93 | issue = 2| pages = 187–205 | doi = 10.1016/j.jctb.2004.09.007 | citeseerx = 10.1.1.572.857 }}</ref>

The number of unlabeled (non-isomorphic) planar graphs on <math>n</math> vertices is between <math>27.2^n</math> and <math>30.06^n</math>.<ref>{{cite journal | last1 = Bonichon | first1 = N. | last2 = Gavoille | first2 = C. | last3 = Hanusse | first3 = N. | last4 = Poulalhon | first4 = D. | last5 = Schaeffer | first5 = G. | year = 2006 | title = Planar Graphs, via Well-Orderly Maps and Trees | journal = Graphs and Combinatorics | volume = 22 | issue = 2| pages = 185–202 | doi=10.1007/s00373-006-0647-2| citeseerx = 10.1.1.106.7456 | s2cid = 22639942 }}</ref>

=== Other results ===

The [[four color theorem]] states that every planar graph is 4-[[graph coloring|colorable]] (i.e., 4-partite).

[[Fáry's theorem]] states that every simple planar graph admits a representation as a [[planar straight-line graph]]. A [[universal point set]] is a set of points such that every planar graph with ''n'' vertices has such an embedding with all vertices in the point set; there exist universal point sets of quadratic size, formed by taking a rectangular subset of the [[integer lattice]]. Every simple outerplanar graph admits an embedding in the plane such that all vertices lie on a fixed circle and all edges are straight line segments that lie inside the disk and don't intersect, so ''n''-vertex [[regular polygon]]s are universal for outerplanar graphs.

[[Scheinerman's conjecture]] (now a theorem) states that every planar graph can be represented as an [[intersection graph]] of [[line segment]]s in the plane.

The [[planar separator theorem]] states that every ''n''-vertex planar graph can be partitioned into two [[Glossary of graph theory#Subgraphs|subgraphs]] of size at most 2''n''/3 by the removal of O({{radic|''n''}}) vertices. As a consequence, planar graphs also have [[treewidth]] and [[branch-width]] O({{radic|''n''}}).

The planar product structure theorem states that every planar graph is a subgraph of the strong [[graph product]] of a graph of treewidth at most 8 and a path.<ref>{{citation
 | last1 = Dujmović | first1 = Vida | author1-link = Vida Dujmović
 | last2 = Joret | first2 = Gwenäel 
 | last3 = Micek | first3 = Piotr
 | last4 = Morin | first4 = Pat | author4-link = Pat Morin
 | last5 = Ueckerdt | first5 = Torsten
 | last6 = Wood | first6 = David R. | author6-link = David Wood (mathematician)
 | title = Planar graphs have bounded queue number
 | journal = Journal of the ACM
 | volume = 67
 | number = 4
 | pages = 22:1–22:38
 | doi = 10.1145/3385731
 | arxiv = 1904.04791
 | year = 2020}}</ref>
This result has been used to show that planar graphs have bounded [[queue number]], bounded [[Thue number|non-repetitive chromatic number]], and [[universal graph]]s of near-linear size.  It also has applications to vertex ranking<ref>{{citation
 | last1 = Bose | first1 = Prosenjit
 | last2 = Dujmović | first2 = Vida | author2-link = Vida Dujmović
 | last3 = Javarsineh | first3 = Mehrnoosh
 | last4 = Morin | first4 = Pat
 | title = Asymptotically optimal vertex ranking of planar graphs
 | arxiv = 2007.06455
 | year = 2020
}}</ref>
and ''p''-centered colouring<ref>{{citation
 | last1 = Dębski | first1 = Michał 
 | last2 = Felsner | first2 = Stefan 
 | last3 = Micek | first3 = Piotr 
 | last4 = Schröder | first4 = Felix
 | title = Improved Bounds for Centered Colorings
 | journal = Advances in Combinatorics 
 | arxiv = 1907.04586
 | year = 2021
| doi = 10.19086/aic.27351 
 | s2cid = 195874032 
 }}</ref>
of planar graphs.

For two planar graphs with ''v'' vertices, it is possible to determine in time O(''v'') whether they are [[graph theory|isomorphic]] or not (see also [[graph isomorphism problem]]).<ref>{{cite book |first1=I. S. |last1=Filotti |first2=Jack N. |last2=Mayer |chapter=A polynomial-time algorithm for determining the isomorphism of graphs of fixed genus |title=Proceedings of the 12th Annual ACM Symposium on Theory of Computing |pages=236–243 |year=1980 |doi=10.1145/800141.804671 |isbn=978-0-89791-017-0|s2cid=16345164 |url=https://hal.inria.fr/inria-00076553/file/RR-0008.pdf }}</ref>

Any planar graph on n nodes has at most 8(n-2) maximal cliques,<ref>Wood, D. R. (2007). On the Maximum Number of Cliques in a Graph. ''Graphs and Combinatorics'', ''23''(3), 337–352. https://doi.org/10.1007/s00373-007-0738-8</ref> which implies that the class of planar graphs is a [[Graphs with few cliques|class with few cliques.]]

==Generalizations==
An [[apex graph]] is a graph that may be made planar by the removal of one vertex, and a ''k''-apex graph is a graph that may be made planar by the removal of at most ''k'' vertices.

A [[1-planar graph]] is a graph that may be drawn in the plane with at most one simple crossing per edge, and a ''k''-planar graph is a graph that may be drawn with at most ''k'' simple crossings per edge.

A [[map graph]] is a graph formed from a set of finitely many simply-connected interior-disjoint regions in the plane by connecting two regions when they share at least one boundary point. When at most three regions meet at a point, the result is a planar graph, but when four or more regions meet at a point, the result can be nonplanar (for example, if one thinks of a circle divided into sectors, with the sectors being the regions, then the corresponding map graph is the complete graph as all the sectors have a common boundary point - the centre point).

A [[toroidal graph]] is a graph that can be embedded without crossings on the [[torus]]. More generally, the [[genus (mathematics)|genus]] of a graph is the minimum genus of a two-dimensional surface into which the graph may be embedded; planar graphs have genus zero and nonplanar toroidal graphs have genus one. Every graph can be embedded without crossings into some (orientable, connected) closed two-dimensional surface (sphere with handles) and thus the genus of a graph is well defined. Obviously, if the graph can be embedded without crossings into a (orientable, connected, closed) surface with genus g, it can be embedded without crossings into all (orientable, connected, closed) surfaces with greater or equal genus. There are also other concepts in graph theory that are called "X genus" with "X" some qualifier; in general these differ from the above defined concept of "genus" without any qualifier. Especially the non-orientable genus of a graph (using non-orientable surfaces in its definition) is different for a general graph from the genus of that graph (using orientable surfaces in its definition).

Any graph may be embedded into three-dimensional space without crossings. In fact, any graph can be drawn without crossings in a two plane setup, where two planes are placed on top of each other and the edges are allowed to "jump up" and "drop down" from one plane to the other at any place (not just at the graph vertices) so that the edges can avoid intersections with other edges. This can be interpreted as saying that it is possible to make any electrical conductor network with a two-sided [[circuit board]] where electrical connection between the sides of the board can be made (as is possible with typical real life circuit boards, with the electrical connections on the top side of the board achieved through pieces of wire and at the bottom side by tracks of copper constructed on to the board itself and electrical connection between the sides of the board achieved through drilling holes, passing the wires through the holes and [[soldering]] them into the tracks); one can also interpret this as saying that in order to build any road network, one only needs just bridges or just tunnels, not both (2 levels is enough, 3 is not needed). Also, in three dimensions the question about drawing the graph without crossings is trivial. However, a three-dimensional analogue of the planar graphs is provided by the [[linkless embedding|linklessly embeddable graphs]], graphs that can be embedded into three-dimensional space in such a way that no two cycles are [[linking number|topologically linked]] with each other. In analogy to Kuratowski's and Wagner's characterizations of the planar graphs as being the graphs that do not contain ''K''<sub>5</sub> or ''K''<sub>3,3</sub> as a minor, the linklessly embeddable graphs may be characterized as the graphs that do not contain as a minor any of the seven graphs in the [[Petersen family]]. In analogy to the characterizations of the outerplanar and planar graphs as being the graphs with [[Colin de Verdière graph invariant]] at most two or three, the linklessly embeddable graphs are the graphs that have Colin de Verdière invariant at most four.

==See also==
* [[Combinatorial map]] a combinatorial object that can encode plane graphs 
* [[Planarization]], a planar graph formed from a drawing with crossings by replacing each crossing point by a new vertex
* [[Thickness (graph theory)]], the smallest number of planar graphs into which the edges of a given graph may be partitioned
* [[Planarity]], a puzzle computer game in which the objective is to embed a planar graph onto a plane
* [[Sprouts (game)]], a pencil-and-paper game where a planar graph subject to certain constraints is constructed as part of the game play
* [[Three utilities problem]], a popular puzzle

==Notes==
<references />

==References==
*{{citation|first=Kazimierz|last=Kuratowski|author-link=Kazimierz Kuratowski|year=1930|url=http://matwbn.icm.edu.pl/ksiazki/fm/fm15/fm15126.pdf|title=Sur le problème des courbes gauches en topologie|language=fr|journal=Fundamenta Mathematicae|volume=15|pages=271–283|doi=10.4064/fm-15-1-271-283|doi-access=free}}.
*{{citation|first=K.|last=Wagner|year=1937|title=Über eine Eigenschaft der ebenen Komplexe|language=de|journal=Mathematische Annalen|volume=114|pages=570–590|doi=10.1007/BF01594196|s2cid=123534907}}.
*{{citation|first1=John M.|last1=Boyer|first2=Wendy J.|last2=Myrvold|author2-link= Wendy Myrvold |year=2005|url=http://jgaa.info/accepted/2004/BoyerMyrvold2004.8.3.pdf|title=On the cutting edge: Simplified O(n) planarity by edge addition|journal=[[Journal of Graph Algorithms and Applications]]|volume=8|issue=3|pages=241–273|doi=10.7155/jgaa.00091|doi-access=free}}.
*{{citation|first1=Brendan|last1=McKay|author-link1=Brendan McKay (mathematician)|first2=Gunnar|last2=Brinkmann|url=http://cs.anu.edu.au/~bdm/plantri/|title=A useful planar graph generator}}.
*{{citation|first1=H.|last1=de Fraysseix|first2=P.|last2=Ossona de Mendez|author2-link = Patrice Ossona de Mendez |first3=P.|last3=Rosenstiehl|year=2006|title=Trémaux trees and planarity|journal=International Journal of Foundations of Computer Science|volume=17|issue=5|pages=1017–1029|doi=10.1142/S0129054106004248|arxiv=math/0610935|s2cid=40107560}}. Special Issue on Graph Drawing.
*{{cite tech report |author1-link=David Bader (computer scientist)|first1=D.A. |last1=Bader |first2=S. |last2=Sreshta  |url=http://www.cc.gatech.edu/~bader/papers/planarity2003.html |archive-url=https://web.archive.org/web/20160316171835/http://www.cc.gatech.edu/~bader/papers/planarity2003.html |archive-date=2016-03-16  |title=A New Parallel Algorithm for Planarity Testing |id=UNM-ECE Technical Report 03-002 |date=October 1, 2003}}
*{{citation|first=Steve|last=Fisk|year=1978|title=A short proof of Chvátal's watchman theorem|journal=Journal of Combinatorial Theory, Series B|volume=24|pages=374|doi=10.1016/0095-8956(78)90059-X|issue=3|doi-access=}}.

==External links==
{{commons category|Planar graphs}}
*[http://jgaa.info/accepted/2004/BoyerMyrvold2004.8.3/planarity.zip Edge Addition Planarity Algorithm Source Code, version 1.0] &mdash; Free C source code for reference implementation of Boyer–Myrvold planarity algorithm, which provides both a combinatorial planar embedder and Kuratowski subgraph isolator. An open source project with free licensing provides the [https://code.google.com/p/planarity/ Edge Addition Planarity Algorithms, current version].
*[http://pigale.sourceforge.net Public Implementation of a Graph Algorithm Library and Editor] &mdash; GPL graph algorithm library including planarity testing, planarity embedder and Kuratowski subgraph exhibition in linear time.
*[http://www.boost.org/doc/libs/1_40_0/libs/graph/doc/planar_graphs.html Boost Graph Library tools for planar graphs], including linear time planarity testing, embedding, Kuratowski subgraph isolation, and straight-line drawing.
*[http://www.cut-the-knot.org/do_you_know/3Utilities.shtml 3 Utilities Puzzle and Planar Graphs]
*[http://ccl.northwestern.edu/netlogo/models/Planarity NetLogo Planarity model] &mdash; NetLogo version of John Tantalo's game

[[Category:Planar graphs| ]]
[[Category:Graph families]]
[[Category:Intersection classes of graphs]]