# Planar graph

Example graphs
PlanarNonplanar

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 K5 (the complete graph on five vertices) or K3,3 (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 K5 or K3,3.

In the Soviet Union, Kuratowski's theorem was known as the Pontryagin-Kuratowski theorem, as its proof was allegedly first given in Pontryagin's unpublished notes. By a long-standing academic tradition, such references are not taken into account in determining priority, so the Russian name of the theorem is not acknowledged internationally.

Here is an example of a graph which doesn't have K5 or K3,3 as its subgraph. However, it has a subgraph that is homeomorphic to K3,3 and is therefore not planar.

Instead of considering subdivisions, Wagner's theorem deals with minors:

A finite graph is planar if and only if it does not have K5 or K3,3 as a minor.

Wagner asked more generally whether any minor-closed class of graphs is determined by a finite set of "forbidden minors". This is now the Robertson-Seymour theorem, proved in a long series of papers. In the language of this theorem, K5 and K3,3 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(v2). The graph K3,3, 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.

### 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, infinitely-large region), then

i.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 ve + 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 3v-6 edges and 2v-4 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 3-connected 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 (K4) is planar but not outerplanar. This is the smallest non-outerplanar 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 K4 (the full graph on 4 vertices) or of K2,3 (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 K4.]

All loopless outerplanar graphs are 3-colorable; this fact features prominently in the simplified proof of Chvátal's art gallery theorem by . A 3-coloring may be found easily by removing a degree-2 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 4-partite, or 4-colorable; this is the graph-theoretical 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.

## 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, UNM-ECE Technical Report 03-002, 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
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.
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
plane is a two-dimensional 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.
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.
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Mazurek Dąbrowskiego   (Polish)
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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.
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
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.
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
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
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 .
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
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.
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
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
Union of Soviet Socialist Republics (abbreviated USSR, Russian: ; tr.
Lev Semenovich Pontryagin (Russian: Лев Семёнович Понтрягин) (3 September 1908 – 3 May 1988) was a Soviet Russian mathematician.
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
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

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.
In mathematics, computing, linguistics, and related disciplines, an algorithm is a finite list of well-defined instructions for accomplishing some task that, given an initial state, will proceed through a well-defined series of successive states, eventually terminating in an
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).
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.
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
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
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