Example graphs
PlanarNonplanar
K5
The complete graph K4 is planar
K3,3

In graph theory, a planar graph is a graph which can be embedded in the plane, i. In the mathematical field of Graph theory, a complete graph is a Simple graph in which every pair of distinct vertices is connected by an In Mathematics and Computer science, graph theory is the study of graphs: mathematical structures used to model pairwise relations between objects In Mathematics and Computer science, a graph is the basic object of study in Graph theory. In Topological graph theory, an embedding of a graph G on a Surface &Sigma is a representation of G on &Sigma in which points e. , it can be drawn on the plane in such a way that its edges may intersect only at their endpoints.

A nonplanar graph is the one which 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 or planar embedding of the graph. A plane graph can be defined as a planar graph with a mapping from every node to a point in 2D space, and from every edge to a plane curve, 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. In mathematics a plane curve is a Curve in a Euclidian plane (cf

It is easily seen that a graph that can be drawn on the plane can be drawn on the sphere as well, and vice versa. "Globose" redirects here See also Globose nucleus. A sphere (from Greek σφαίρα - sphaira, "globe

The equivalence class of topologically equivalent drawings on the sphere is called a planar map. In Mathematics, given a set X and an Equivalence relation ~ on X, the equivalence class of an element a in X In Mathematics, two functions are said to be topologically conjugate to one another if there exists a Homeomorphism that will conjugate the one into the other Although a plane graph has an external or unbounded face, none of the faces of a planar map have a particular status.

A generalization of planar graphs are graphs which can be drawn on a surface of a given genus. In Mathematics, genus has a few different but closely related meanings Topology Orientable surface In this terminology, planar graphs have graph genus 0, since the plane (and the sphere) are surfaces of genus 0. In Topological graph theory, an embedding of a graph G on a Surface &Sigma is a representation of G on &Sigma in which points See "graph embedding" for other related topics. In Topological graph theory, an embedding of a graph G on a Surface &Sigma is a representation of G on &Sigma in which points

## 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). Poland (Polska officially the Republic of Poland Kazimierz Kuratowski ( Warsaw, February 2, 1896 &ndash June 18, 1980) was a Polish Mathematician and Logician 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 In Mathematics and Computer science, a graph is the basic object of study in Graph theory. Graph theory is a growing area in mathematical research and has a large specialized vocabulary In Graph theory, two graphs G and G' are homeomorphic if there is an Isomorphism from some subdivision of G to some In the mathematical field of Graph theory, a complete graph is a Simple graph in which every pair of distinct vertices is connected by an For other uses see Vertex. In Graph theory, a vertex (plural vertices) or node is the fundamental unit out In the Mathematical field of Graph theory, a complete bipartite graph or biclique is a special kind of Bipartite graph where every

A subdivision of a graph results from inserting vertices into edges (for example, changing an edge •——• to •—•—•) zero or more times. In Graph theory, two graphs G and G' are homeomorphic if there is an Isomorphism from some subdivision of G to some 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 Graph theory, two graphs G and G' are homeomorphic if there is an Isomorphism from some subdivision of G to some

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. The Union of Soviet Socialist Republics (USSR was a constitutionally Socialist state that existed in Eurasia from 1922 to 1991 Lev Semenovich Pontryagin ( Russian Лев Семёнович Понтрягин ( 3 September 1908 &ndash 3 May 1988) was a 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. In Graph theory, a graph H is called a minor of the graph G if H is isomorphic to a graph that can be obtained by zero or more edge In Graph theory, a graph H is called a minor of the graph G if H is isomorphic to a graph that can be obtained by zero or more edge

Klaus Wagner asked more generally whether any minor-closed class of graphs is determined by a finite set of "forbidden minors". Klaus Wagner (March 31 1910 – February 6 2000 was a German mathematician In Graph theory, the Robertson–Seymour theorem (also called the graph minor theorem) states that the minor ordering on the finite undirected graphs is This is now the Robertson-Seymour theorem, proved in a long series of papers. In Graph theory, the Robertson–Seymour theorem (also called the graph minor theorem) states that the minor ordering on the finite undirected graphs is In the language of this theorem, K5 and K3,3 are the forbidden children for the class of finite planar graphs. A language is a dynamic set of visual auditory or tactile Symbols of Communication and the elements used to manipulate them

## 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 may be planar or not (see planarity testing). In Mathematics, Computing, Linguistics and related subjects an algorithm is a sequence of finite instructions often used for Calculation In mathematics big O notation (so called because it uses the symbol O) describes the limiting behavior of a function for very small or very large arguments In Graph theory, the planarity testing problem asks whether given a graph that graph is a Planar graph (can be drawn in the plane without edge intersections

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). In Mathematics, a dense graph is a graph in which the number of edges is close to the maximal number of edges 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 (see also graph isomorphism problem). In Mathematics and Computer science, graph theory is the study of graphs: mathematical structures used to model pairwise relations between objects In Graph theory, an isomorphism of graphs G and H is a Bijection between the vertex sets of G and H [1]

• 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;
• Fraysseix-Rosenstiehl's planarity criterion gives a characterization based on the existence of a bipartition of the cotree edges of a depth-first search tree. In Mathematics, a dual graph of a given Planar graph G is a graph which has a vertex for each plane region of G, and an edge for each edge 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 The binary cycle space In Graph theory, certain Vector spaces over the two-element field Z 2 are associated with an undirected In Graph theory, a branch of Mathematics, Fraysseix–Rosenstiehl's planarity criterion is a characterization of planarity based on the properties of the It is central to the left-right 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. In Mathematics, Schnyder's theorem in Graph theory is a planarity characterization for graphs in termsof the Order dimension of their incidence In Mathematics, the dimension of a Partially ordered set (poset is the smallest number of Total orders the intersection of which gives rise to the partial Colin de Verdière's invariant is a graph parameter \mu(G for any graph G which has been introduced by Colin de Verdière in 1990 motivated

### 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

ve + f = 2

i. In Mathematics and Computer science, graph theory is the study of graphs: mathematical structures used to model pairwise relations between objects Graph theory is a growing area in mathematical research and has a large specialized vocabulary e. the Euler characteristic is 2. In Mathematics, and more specifically in Algebraic topology and Polyhedral combinatorics, the Euler characteristic is a Topological invariant 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. In Graph theory, a tree is a graph in which any two vertices are connected by exactly one path. Cycle in Graph theory and Computer science has several meanings A closed walk with repeated vertices allowed 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. In Mathematics and Computer science, connectivity is one of the basic concepts of Graph theory. In Mathematics and Computer science, a graph is the basic object of study in Graph theory.

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. What is a polyhedron? We can at least say that a polyhedron is built up from different kinds of element or entity each associated with a different number of dimensions 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. A tetrahedron (plural tetrahedra) is a Polyhedron composed of four triangular faces three of which meet at each vertex. 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. Ernst Steinitz ( June 13, 1871 – September 29 1928) was a German Mathematician. In Euclidean space, an object is convex if for every pair of points within the object every point on the Straight line segment that joins them is also within the In Mathematics and Computer science, connectivity is one of the basic concepts of Graph theory.

### 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. Circles are simple Shapes of Euclidean geometry consisting of those points in a plane which are at a constant Distance, called the In Geometry, a disk (also spelled disc) is the region in a plane bounded by a Circle. 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. In Graph theory, a tree is a graph in which any two vertices are connected by exactly one path.

An outerplanar graph without loops (edges with coinciding endvertices) has a vertex of degree at most 2.

All loopless outerplanar graphs are 3-colorable; this fact features prominently in the simplified proof of Chvátal's art gallery theorem by Fisk (1978). Václav (Vašek Chvátal (born 1946 (ˈvaːt͡slaf ˈxvaːtal is a professor in the Department of Computer Science and Software Engineering at Concordia University in 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.

#### k-outerplanar graphs

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

## 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. In Graph theory, graph coloring is a special case of Graph labeling; it is an assignment of labels traditionally called "colors" to elements of a The four color theorem (also known as the four color map theorem) states that given any plane separated into regions such as a political map of the states of a country

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. Fáry's theorem states that any simple Planar graph can be drawn without crossings so that its edges are straight Line segments That is the ability to 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.

A planar graph and its dual

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. In Mathematics, a dual graph of a given Planar graph G is a graph which has a vertex for each plane region of G, and an edge for each edge In Mathematics, a dual graph of a given Planar graph G is a graph which has a vertex for each plane region of G, and an edge for each edge 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. "Globose" redirects here See also Globose nucleus. A sphere (from Greek σφαίρα - sphaira, "globe 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.

A Euclidean graph is a graph in which the vertices represent points in the plane, and the edges are assigned lengths equal to the Euclidean distance between those points; see Geometric graph theory. In Mathematics, a geometric graph is a graph in which the vertices or edges are associated with geometric objects or configurations