Planar graph
Example graphs  

Planar  Nonplanar 
In graph theory, a planar graph is a graph that can be drawn so that no edges intersect (or that can be embedded) in the plane. A nonplanar graph cannot be drawn in the plane without edge intersections.
A planar graph already drawn in the plane without edge intersections is called a plane graph. A plane graph can be defined as a planar graph with a mapping from every node to a position in 2D space, and from every edge to a plane curve, such that each curve has two extreme points, which coincide with the positions of its end nodes, and all curves are disjoint except on their extreme points.
The equivalence class of topologically equivalent drawings on the sphere is called a planar map. Although a plane graph has an external or unbounded face, none of the faces of a planar map has a particular status.
Kuratowski's and Wagner's theorems
The Polish mathematician Kazimierz Kuratowski provided a characterization of planar graphs in terms of forbidden graphs, now known as Kuratowski's theorem: A finite graph is planar if and only if it does not contain a subgraph that is a subdivision of K_{5} (the complete graph on five vertices) or K_{3,3} (complete bipartite graph on six vertices, three of which connect to each of the other three).
A subdivision of a graph results from inserting vertices into edges (for example, changing an edge •——• to •—•—•) and repeating this zero or more times. Equivalent formulations of this theorem, also known as "Theorem P" include
 A finite graph is planar if and only if it does not contain a subgraph that is homeomorphic to K_{5} or K_{3,3}.
In the Soviet Union, Kuratowski's theorem was known as the PontryaginKuratowski theorem, as its proof was allegedly first given in Pontryagin's unpublished notes. By a longstanding academic tradition, such references are not taken into account in determining priority, so the Russian name of the theorem is not acknowledged internationally.
Instead of considering subdivisions, Wagner's theorem deals with minors:
 A finite graph is planar if and only if it does not have K_{5} or K_{3,3} as a minor.
Wagner asked more generally whether any minorclosed class of graphs is determined by a finite set of "forbidden minors". This is now the RobertsonSeymour theorem, proved in a long series of papers. In the language of this theorem, K_{5} and K_{3,3} are the forbidden children for the class of finite planar graphs.
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 is planar or not.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(v^{2}). 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.
For two planar graphs with v vertices, it is possible to determine in time O(v) whether they are isomorphic or not.
 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;
 FraysseixRosenstiehl's planarity criterion gives a characterization based on the existence of a bipartition of the cotree edges of a depthfirst search tree. It is central to the leftright 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.
Euler's formula
Euler's formula states that if a finite, connected, planar graph is drawn in the plane without any edge intersections, and v is the number of vertices, e is the number of edges and f is the number of faces (regions bounded by edges, including the outer, infinitelylarge region), theni.e. the Euler characteristic is 2. As an illustration, in the first planar graph given above, we have v=6, e=7 and f=3. If the second graph is redrawn without edge intersections, we get v=4, e=6 and f=4. Euler's formula can be proven as follows: if the graph isn't a tree, then remove an edge which completes a cycle. This lowers both e and f by one, leaving v − e + f constant. Repeat until you arrive at a tree; trees have v = e + 1 and f = 1, yielding v  e + f = 2.
In a finite, connected, 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; using Euler's formula, one can then show that these graphs are sparse in the sense that e ≤ 3v  6 if v ≥ 3.
A simple graph is called maximal planar if it is planar but adding any edge would destroy that property. All faces (even the outer one) are then bounded by three edges, explaining the alternative term triangular for these graphs. If a triangular graph has v vertices with v > 2, then it has precisely 3v6 edges and 2v4 faces.
Note that Euler's formula is also valid for simple polyhedra. This is no coincidence: every simple polyhedron can be turned into a connected, simple, planar graph by using the polyhedron's vertices as vertices of the graph and the polyhedron's edges as edges of the graph. The faces of the resulting planar graph then correspond to the faces of the polyhedron. For example, the second planar graph shown above corresponds to a tetrahedron. Not every connected, simple, planar graph belongs to a simple polyhedron in this fashion: the trees do not, for example. A theorem of Ernst Steinitz says that the planar graphs formed from convex polyhedra (equivalently: those formed from simple polyhedra) are precisely the finite 3connected simple planar graphs.
Outerplanar graphs
A graph is called outerplanar if it has an embedding in the plane such that the vertices lie on a fixed circle and the edges lie inside the disk of the circle and don't intersect. Equivalently, there is some face that includes every vertex. Every outerplanar graph is planar, but the converse is not true: the second example graph shown above (K_{4}) is planar but not outerplanar. This is the smallest nonouterplanar graph: a theorem similar to Kuratowski's states that a finite graph is outerplanar if and only if it does not contain a subgraph that is an expansion of K_{4} (the full graph on 4 vertices) or of K_{2,3} (five vertices, 2 of which connected to each of the other three for a total of 6 edges).Properties of outerplanar graphs
All finite or countably infinite trees are outerplanar and hence planar.An outerplanar graph has a vertex of degree at most 2 or a looped vertex of degree 4. [otherwise there must be at least 4 vertices of degree at least 3 or looped vertices of degree at least 5; such a graph can be retracted to a K_{4}.]
All loopless outerplanar graphs are 3colorable; this fact features prominently in the simplified proof of Chvátal's art gallery theorem by . A 3coloring may be found easily by removing a degree2 vertex, coloring the remaining graph recursively, and adding back the removed vertex with a color different from its two neighbors.
Other facts and definitions
Every planar graph without loops is 4partite, or 4colorable; this is the graphtheoretical formulation of the four color theorem.Fáry's theorem states that every simple planar graph admits an embedding in the plane such that all edges are straight line segments which don't intersect. Similarly, 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.
Given an embedding G of a (not necessarily simple) planar graph in the plane without edge intersections, we construct the dual graph G* as follows: we choose one vertex in each face of G (including the outer face) and for each edge e in G we introduce a new edge in G* connecting the two vertices in G* corresponding to the two faces in G that meet at e. Furthermore, this edge is drawn so that it crosses e exactly once and that no other edge of G or G* is intersected. Then G* is again the embedding of a (not necessarily simple) planar graph; it has as many edges as G, as many vertices as G has faces and as many faces as G has vertices. The term "dual" is justified by the fact that G** = G; here the equality is the equivalence of embeddings on the sphere. If G is the planar graph corresponding to a convex polyhedron, then 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.
External links
 Edge Addition Planarity Algorithm Source Code — Free C source code for reference implementation of BoyerMyrvold planarity algorithm, which provides both a combinatorial planar embedder and Kuratowski subgraph isolator.
 Public Implementation of a Graph Algorithm Library and Editor — GPL graph algorithm library including planarity testing, planarity embedder and Kuratowski subgraph exhibition in linear time.
 3 Utilities Puzzle and Planar Graphs
 Planarity — A puzzle game created by John Tantalo.
References

id="CITEREFKuratowski1930">Kuratowski, Kazimierz (1930), "Sur le problème des courbes gauches en topologie", Fund. Math. 15: 271–283, <[1].

id="CITEREFWagner1937">Wagner, K. (1937), "Über eine Eigenschaft der ebenen Komplexe", Math. Ann. 114: 570–590.

id="CITEREFBoyerMyrvold2005">Boyer, John M. & Wendy J. Myrvold (2005), "On the cutting edge: Simplified O(n) planarity by edge addition", Journal of Graph Algorithms and Applications 8 (3): 241–273, <[2].

id="CITEREFMcKayBrinkmann">McKay, Brendan & Gunnar Brinkmann, A useful planar graph generator, <[3].

id="CITEREFde FraysseixOssona de MendezRosenstiehl2006">de Fraysseix, H.; P. Ossona de Mendez & P. Rosenstiehl (2006), "Trémaux trees and planarity", International Journal of Foundations of Computer Science 17 (5): 1017–1029. Special Issue on Graph Drawing. doi:10.1142/S0129054106004248
 D.A. Bader and S. Sreshta, A New Parallel Algorithm for Planarity Testing, UNMECE Technical Report 03002, October 1, 2003.

id="CITEREFFisk1978">Fisk, Steve (1978), "A short proof of Chvátal's watchman theorem", J. Comb. Theory, Ser. B 24: 374.
graph theory is the study of graphs; mathematical structures used to model pairwise relations between objects from a certain collection. A "graph" in this context refers to a collection of vertices or 'nodes' and a collection of edges
..... Click the link for more information.graph is the basic object of study in graph theory. Informally speaking, a graph is a set of objects called points, nodes, or vertices connected by links called lines or edges.
..... Click the link for more information.embedding (or imbedding) is one instance of some mathematical object contained within another instance, such as a group that is a subgroup.Abstractly or categorically
An abstract embedding
..... Click the link for more information.plane is a twodimensional manifold or surface that is perfectly flat. Informally it can be thought of as an infinitely vast and infinitesimally thin sheet oriented in some space.
..... Click the link for more information.In mathematics, a plane curve is a curve in a Euclidian plane (cf. space curve). The most frequently studied cases are smooth plane curves (including piecewise smooth plane curves), and algebraic plane curves.
..... Click the link for more information.Motto
none^{1}
Anthem
Mazurek Dąbrowskiego (Polish)
Dąbrowski's Mazurek
..... Click the link for more information.Kazimierz Kuratowski (Warsaw, February 2, 1896 — June 18, 1980) was a Polish mathematician and logician.Biography
Kuratowski became a professor of mathematics in 1927 at the Lwów Polytechnic in Lwów, Poland, and from 1934 at Warsaw University.
..... Click the link for more information.A forbidden graph characterization is a method of specifying or describing a family of graphs whereby a graph belongs to the family in question if and only if for the graph in question certain graphs, called forbidden graphs
..... Click the link for more information.graph is the basic object of study in graph theory. Informally speaking, a graph is a set of objects called points, nodes, or vertices connected by links called lines or edges.
..... Click the link for more information.“Iff” redirects here. For other uses, see IFF.If and only if, in logic and fields that rely on it such as mathematics and philosophy, is a logical connective between statements which means that the truth of either one of the statements
..... Click the link for more information.In graph theory, two graphs and are homeomorphic if there is an isomorphism from some subdivision of to some subdivision of . If the edges of a graph are thought of as lines drawn from one vertex to another (as they are usually depicted in illustrations), then two
..... Click the link for more information.In the mathematical field of graph theory, a complete graph is a simple graph where an edge connects every pair of distinct vertices. The complete graph on vertices has vertices and edges, and is denoted by . It is a regular graph of degree .
..... Click the link for more information.vertex (plural vertices) or node is the fundamental unit out of which graphs are formed: an undirected graph consists of a set of vertices and a set of edges (unordered pairs of vertices), while a directed graph consists of a set of vertices and a set of arcs (ordered
..... Click the link for more information.In the mathematical field of graph theory, a complete bipartite graph or biclique is a special kind of bipartite graph where every vertex of the first set is connected to every vertex of the second set.
..... Click the link for more information.In graph theory, two graphs and are homeomorphic if there is an isomorphism from some subdivision of to some subdivision of . If the edges of a graph are thought of as lines drawn from one vertex to another (as they are usually depicted in illustrations), then two
..... Click the link for more information.In graph theory, two graphs and are homeomorphic if there is an isomorphism from some subdivision of to some subdivision of . If the edges of a graph are thought of as lines drawn from one vertex to another (as they are usually depicted in illustrations), then two
..... Click the link for more information.Union of Soviet Socialist Republics (abbreviated USSR, Russian: (help info ) ; tr.
..... Click the link for more information.Lev Semenovich Pontryagin (Russian: Лев Семёнович Понтрягин) (3 September 1908 – 3 May 1988) was a Soviet Russian mathematician.
..... Click the link for more information.minor of the graph G if H is isomorphic to a graph that can be obtained by zero or more edge contractions on a subgraph of G. Edge contraction is the process of removing an edge and combining its two endpoints into a single node (since the edge is first
..... Click the link for more information.minor of the graph G if H is isomorphic to a graph that can be obtained by zero or more edge contractions on a subgraph of G. Edge contraction is the process of removing an edge and combining its two endpoints into a single node (since the edge is first
..... Click the link for more information.See Language (journal) for the linguistics journal.
A language is a system of symbols and the rules used to manipulate them. Language can also refer to the use of such systems as a general phenomenon.
..... Click the link for more information.In mathematics, computing, linguistics, and related disciplines, an algorithm is a finite list of welldefined instructions for accomplishing some task that, given an initial state, will proceed through a welldefined series of successive states, eventually terminating in an
..... Click the link for more information.In computational complexity theory, big O notation is often used to describe how the size of the input data affects an algorithm's usage of computational resources (usually running time or memory).
..... Click the link for more information.In mathematics, a dense graph is a graph in which the number of edges is close to the maximal number of edges. The opposite, a graph with only a few edges, is a sparse graph.
The distinction between sparse and dense graphs is rather vague.
..... Click the link for more information.graph theory is the study of graphs; mathematical structures used to model pairwise relations between objects from a certain collection. A "graph" in this context refers to a collection of vertices or 'nodes' and a collection of edges
..... Click the link for more information.dual graph of a given planar graph G has a vertex for each plane region of G, and an edge for each edge joining two neighboring regions. The term "dual" is used because this property is symmetric, meaning that if G is a dual of H, then H
..... Click the link for more information.In graph theory, Mac Lane's planarity criterion is a characterisation of planar graphs in terms of their cycle spaces. It states that a finite graph G is planar if and only if the cycle space C(G
..... Click the link for more information.Z_{2} are associated with an undirected graph; this allows one to use the tools of linear algebra to study graphs.
Let G be a finite simple undirected graph with edge set E.
..... Click the link for more information.In mathematics, FraysseixRosenstiehl's planarity criterion in graph theory is based on the properties of the tree defined by a depthfirst search. Considering any depthfirst search of a graph G
..... Click the link for more information.In mathematics, Schnyder's theorem in graph theory is a planarity characterization for graphs in terms of the order dimension of their incidence posets.
The incidence poset of a graph G with vertex set V and edge set E
..... Click the link for more information.

id="CITEREFde FraysseixOssona de MendezRosenstiehl2006">de Fraysseix, H.; P. Ossona de Mendez & P. Rosenstiehl (2006), "Trémaux trees and planarity", International Journal of Foundations of Computer Science 17 (5): 1017–1029. Special Issue on Graph Drawing. doi:10.1142/S0129054106004248

id="CITEREFMcKayBrinkmann">McKay, Brendan & Gunnar Brinkmann, A useful planar graph generator, <[3].

id="CITEREFBoyerMyrvold2005">Boyer, John M. & Wendy J. Myrvold (2005), "On the cutting edge: Simplified O(n) planarity by edge addition", Journal of Graph Algorithms and Applications 8 (3): 241–273, <[2].

id="CITEREFWagner1937">Wagner, K. (1937), "Über eine Eigenschaft der ebenen Komplexe", Math. Ann. 114: 570–590.
This article is copied from an article on Wikipedia.org  the free encyclopedia created and edited by online user community. The text was not checked or edited by anyone on our staff. Although the vast majority of the wikipedia encyclopedia articles provide accurate and timely information please do not assume the accuracy of any particular article. This article is distributed under the terms of GNU Free Documentation License.