Planar graph - Wikipedia, the free encyclopedia Page 1 of 7 Planar graph From Wikipedia, the free encyclopedia In graph theory, a planar graph is a graph that can be Example graphs embedded in the plane, i.e., it can be drawn on the plane in such a way that its edges intersect only at their endpoints. In other Planar Nonplanar words, it can be drawn in such a way that no edges cross each other. A planar graph already drawn in the plane without edge intersections is called a plane graph or planar embedding of Butterfly graph 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 K5 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. Plane graphs can be encoded by combinatorial maps. 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. The equivalence class of topologically equivalent drawings on K3,3 the sphere is called a planar map . Although a plane graph has The complete graph external unbounded an or face, none of the faces of a planar K4 is 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 this terminology, planar graphs have graph genus 0, since the plane (and the sphere) are surfaces of genus 0. See "graph embedding" for other related topics. Contents 1 Kuratowski's and Wagner's theorems 2 Other planarity criteria 2.1 Euler's formula 3 Related families of graphs 4 Other facts and definitions 5 Applications 6 Notes 7 References 8 External links 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 http://en.wikipedia.org/wiki/Planar_graph 5/23/2011 Planar graph - Wikipedia, the free encyclopedia Page 2 of 7 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 •—•—•) 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. 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. 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. Klaus 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 minors 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 may be planar or not (see planarity testing ). http://en.wikipedia.org/wiki/Planar_graph 5/23/2011 Planar graph - Wikipedia, the free encyclopedia Page 3 of 7 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 ≤ 3 v − 6; Theorem 2. If v > 3 and there are no cycles of length 3, then e ≤ 2 v − 4. In this sense, planar graphs are sparse graphs, in that they have only O( v) edges, asymptotically smaller than the maximum O An animation showing that the Petersen graph contains a minor isomorphic to the 2 K3,3 graph (v ). 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). [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. 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. 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 v − e + f = 2 . As an illustration, in the butterfly graph given above, v = 5, e = 6 and f = 3. If the second graph is http://en.wikipedia.org/wiki/Planar_graph 5/23/2011 Planar graph - Wikipedia, the free encyclopedia Page 4 of 7 redrawn without edge intersections, it has v = 4, e = 6 and f = 4. In general, if the property holds for all planar graphs of f faces, any change to the graph that creates an additional face while keeping the graph planar would keep v − e + f an invariant. Since the property holds for all graphs with f = 2, by mathematical induction it holds for all cases. Euler's formula can also 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 the remaining graph is a tree; trees have v = e + 1 and f = 1, yielding v − e + f = 2. i.e. the Euler characteristic is 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 ≤ 3 v − 6 if v ≥ 3. 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 (even the outer one) are then bounded by three edges, explaining the alternative terms triangular and triangulated for these graphs. If a triangular graph has v vertices with v > 2, then it has precisely 3 v − 6 edges and 2v − 4 faces. Euler's formula is also valid for simple polyhedra. This is no coincidence: every simple polyhedron can be turned into a The Goldner–Harary graph is maximal connected, simple, planar graph by using the polyhedron's planar. All its faces are bounded by three vertices as vertices of the graph and the polyhedron's edges as edges. 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. Steinitz's theorem says that the polyhedral graphs formed from convex polyhedra (equivalently: those formed from simple polyhedra) are precisely the finite 3-connected simple planar graphs. Related families of graphs Outerplanar graphs 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: K4 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 K4 or of K2,3 . 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 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.