Planar graph

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In graph theory, a planar graph is a graph that can be drawn (mathematicians say "can be embedded in the plane") so that no edges intersect. A nonplanar graph cannot be drawn without edge intersections.

A planar graph already drawn in the plane 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.

[edit] Kuratowski's and Wagner's theorems

The Polish mathematician Kazimierz Kuratowski provided a characterization of planar 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 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 K₅ or K_3,3.

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theorem was known as the Pontryagin-Kuratowski theorem, as its proof was allegedly first given in Pontryagin's unpublished notes.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.

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 K_3,3 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 K_3,3 are the forbidden minors for the class of finite planar graphs.

[edit] 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 n vertices and e edges, the following simple planarity criteria hold:

Theorem 1. If n ≥ 3 then e ≤ 3n - 6

Theorem 2. If n > 3 and there are no cycles of length 3, then e ≤ 2n - 4

In this sense, planar graphs are sparse graphs, in that they have only O(n) edges, asymptotically smaller than the maximum O(n²). The graph K_3,3, for example, has 6 vertices, 9 edges, and no cycles of length 3. Therefore, by Theorem 2, it can not be planar. Note that these theorems are if, not if and only if, 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 must be used.

For two planar graphs with n vertices, it is possible to determine in time O(n) 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;
• 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. 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.

[edit] 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

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

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 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 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.

[edit] 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 K_2,3 (five vertices, 2 of which connected to each of the other three for a total of 6 edges).

All finite or countably infinite trees are outerplanar and hence planar.

All outerplanar graphs are 3-colorable.

[edit] 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.

Image:Dual graphs.svg 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.

[edit] External links

• Edge Addition Planarity Algorithm Source Code — Free C source code for reference implementation of Boyer-Myrvold 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 — An interesting puzzle game created by John Tantalo. Warning: addictive.

[edit] References

• Kazimierz Kuratowski (1930), 'Sur le problème des courbes gauches en topologie', Fund. Math., 15, pp. 271-283.
• K. Wagner (1937), 'Über eine Eigenschaft der ebenen Komplexe', Math. Ann. 114, pp. 570–590.
• John M. Boyer and Wendy J. Myrvold (2005), 'On the cutting edge: Simplified O(n) planarity by edge addition'. Journal of Graph Algorithms and Applications, vol. 8 no. 3, pp. 241-273.
• Brendan McKay & Gunnar Brinkmann, A useful planar graph generator
• H. de Fraysseix, P. Ossona de Mendez & P. Rosenstiehl (2006). 'Trémaux trees and planarity'. International Journal of Foundations of Computer Science, vol. 17 no. 5, pp. 1017-1029. Special Issue on Graph Drawing. doi:10.1142/S0129054106004248cs:Rovinný graf

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