Example graphs | |
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Planar | Nonplanar |
![]() Butterfly graph | ![]() Complete graphK5 |
![]() Complete graph K4 | ![]() Utility graphK3,3 |
Ingraph theory, aplanar graph is agraph that can beembedded in theplane, i.e., it can be drawn on the plane in such a way that its edges intersect only at their endpoints. In other words, it can be drawn in such a way that no edges cross each other.[1][2] Such a drawing is called aplane graph, or aplanar embedding of the graph. A plane graph can be defined as a planar graph with a mapping from every node to a point on a plane, and from every edge to aplane curve on that plane, 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.
Every graph that can be drawn on a plane can be drawn on thesphere as well, and vice versa, by means ofstereographic projection.
Plane graphs can be encoded bycombinatorial maps orrotation systems.
Anequivalence class oftopologically equivalent drawings on the sphere, usually with additional assumptions such as the absence ofisthmuses, is called aplanar map. Although a plane graph has anexternal orunboundedface, none of the faces of a planar map has a particular status.
Planar graphs generalize to graphs drawable on a surface of a givengenus. In this terminology, planar graphs havegenus 0, since the plane (and the sphere) are surfaces of genus 0. See "graph embedding" for other related topics.
ThePolish mathematicianKazimierz Kuratowski provided a characterization of planar graphs in terms offorbidden graphs, now known asKuratowski's theorem:
Asubdivision of a graph results from inserting vertices into edges (for example, changing an edge• —— • to • — • — • ) zero or more times.
Instead of considering subdivisions,Wagner's theorem deals withminors:
Aminor of a graph results from taking a subgraph and repeatedly contracting an edge into a vertex, with each neighbor of the original end-vertices becoming a neighbor of the new vertex.
Klaus Wagner asked more generally whether any minor-closed class of graphs is determined by a finite set of "forbidden minors". This is now theRobertson–Seymour theorem, proved in a long series of papers. In the language of this theorem,K5 andK3,3 are the forbidden minors for the class of finite planar graphs.
In practice, it is difficult to use Kuratowski's criterion to quickly decide whether a given graph is planar. However, there exist fastalgorithms for this problem: for a graph withn vertices, it is possible to determine in timeO(n) (linear time) whether the graph may be planar or not (seeplanarity testing).
For a simple, connected, planar graph withv vertices ande edges andf faces, the following simple conditions hold forv ≥ 3:
In this sense, planar graphs aresparse graphs, in that they have onlyO(v) edges, asymptotically smaller than the maximumO(v2). The graphK3,3, for example, has 6 vertices, 9 edges, and no cycles of length 3. Therefore, by Theorem 2, it cannot be planar. 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.
Euler's formula states that if a finite,connected, planar graph is drawn in the plane without any edge intersections, andv is the number of vertices,e is the number of edges andf is the number of faces (regions bounded by edges, including the outer, infinitely large region), then
As an illustration, in thebutterfly graph given above,v = 5,e = 6 andf = 3. In general, if the property holds for all planar graphs off faces, any change to the graph that creates an additional face while keeping the graph planar would keepv –e +f an invariant. Since the property holds for all graphs withf = 2, bymathematical induction it holds for all cases. Euler's formula can also be proved as follows: if the graph isn't atree, then remove an edge which completes acycle. This lowers bothe andf by one, leavingv –e +f constant. Repeat until the remaining graph is a tree; trees havev =e + 1 andf = 1, yieldingv –e +f = 2, i. e., theEuler 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, so3f <= 2e; using Euler's formula, one can then show that these graphs aresparse in the sense that ifv ≥ 3:
Euler's formula is also valid forconvex polyhedra. This is no coincidence: every convex polyhedron can be turned into a connected, simple, planar graph by using theSchlegel diagram of the polyhedron, aperspective projection of the polyhedron onto a plane with the center of perspective chosen near the center of one of the polyhedron's faces. Not every planar graph corresponds to a convex polyhedron in this way: the trees do not, for example.Steinitz's theorem says that thepolyhedral graphs formed from convex polyhedra are precisely the finite3-connected simple planar graphs. More generally, Euler's formula applies to any polyhedron whose faces aresimple polygons that form a surfacetopologically equivalent to a sphere, regardless of its convexity.
Connected planar graphs with more than one edge obey the inequality2e ≥ 3f, because each face has at least three face-edge incidences and each edge contributes exactly two incidences. It follows via algebraic transformations of this inequality with Euler's formulav –e +f = 2 that for finite planar graphs the average degree is strictly less than 6. Graphs with higher average degree cannot be planar.
We say that two circles drawn in a planekiss (orosculate) whenever they intersect in exactly one point. A "coin graph" is a graph formed by a set of circles, no two of which have overlapping interiors, by making a vertex for each circle and an edge for each pair of circles that kiss. Thecircle packing theorem, first proved byPaul Koebe in 1936, states that a graph is planar if and only if it is a coin graph.
This result provides an easy proof ofFáry's theorem, that every simple planar graph can be embedded in the plane in such a way that its edges are straightline segments that do not cross each other. If one places each vertex of the graph at the center of the corresponding circle in a coin graph representation, then the line segments between centers of kissing circles do not cross any of the other edges.
Themeshedness coefficient or densityD of a planar graph, or network, is the ratio of the numberf – 1 of bounded faces (the same as thecircuit rank of the graph, byMac Lane's planarity criterion) by its maximal possible values2v – 5 for a graph withv vertices:
The density obeys0 ≤D ≤ 1, withD = 0 for a completely sparse planar graph (a tree), andD = 1 for a completely dense (maximal) planar graph.[3]
Given an embeddingG of a (not necessarily simple) connected graph in the plane without edge intersections, we construct thedual graphG* as follows: we choose one vertex in each face ofG (including the outer face) and for each edgee inG we introduce a new edge inG* connecting the two vertices inG* corresponding to the two faces inG that meet ate. Furthermore, this edge is drawn so that it crossese exactly once and that no other edge ofG orG* is intersected. ThenG* is again the embedding of a (not necessarily simple) planar graph; it has as many edges asG, as many vertices asG has faces and as many faces asG has vertices. The term "dual" is justified by the fact thatG** =G; here the equality is the equivalence of embeddings on thesphere. IfG is the planar graph corresponding to a convex polyhedron, thenG* 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.
While the dual constructed for a particular embedding is unique (up toisomorphism), graphs may have different (i.e. non-isomorphic) duals, obtained from different (i.e. non-homeomorphic) embeddings.
A simple graph is calledmaximal planar if it is planar but adding any edge (on the given vertex set) would destroy that property. All faces (including the outer one) are then bounded by three edges, explaining the alternative termplane triangulation (which technically means a plane drawing of the graph). The alternative names "triangular graph"[4] or "triangulated graph"[5] have also been used, but are ambiguous, as they more commonly refer to theline graph of acomplete graph and to thechordal graphs respectively. Every maximal planar graph on more than 3 vertices is at least 3-connected.[6]
If a maximal planar graph hasv vertices withv > 2, then it has precisely3v – 6 edges and2v – 4 faces.
Apollonian networks are the maximal planar graphs formed by repeatedly splitting triangular faces into triples of smaller triangles. Equivalently, they are the planar3-trees.
Strangulated graphs are the graphs in which everyperipheral cycle is a triangle. In a maximal planar graph (or more generally a polyhedral graph) the peripheral cycles are the faces, so maximal planar graphs are strangulated. The strangulated graphs include also thechordal graphs, and are exactly the graphs that can be formed byclique-sums (without deleting edges) ofcomplete graphs and maximal planar graphs.[7]
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 ofK4 or ofK2,3. The above is a direct corollary of the fact that a graphG is outerplanar if the graph formed fromG by adding a new vertex, with edges connecting it to all the other vertices, is a planar graph.[8]
A 1-outerplanar embedding of a graph is the same as an outerplanar embedding. Fork > 1 a planar embedding isk-outerplanar if removing the vertices on the outer face results in a(k – 1)-outerplanar embedding. A graph isk-outerplanar if it has ak-outerplanar embedding.
AHalin graph is a graph formed from an undirected plane tree (with no degree-two nodes) by connecting its leaves into a cycle, in the order given by the plane embedding of the tree. Equivalently, it is apolyhedral graph in which one face is adjacent to all the others. Every Halin graph is planar. Like outerplanar graphs, Halin graphs have lowtreewidth, making many algorithmic problems on them more easily solved than in unrestricted planar graphs.[9]
Anupward planar graph is adirected acyclic graph that can be drawn in the plane with its edges as non-crossing curves that are consistently oriented in an upward direction. Not every planar directed acyclic graph is upward planar, and it isNP-complete to test whether a given graph is upward planar.
A planar graph is said to beconvex if all of its faces (including the outer face) areconvex polygons. Not all planar graphs have a convex embedding (e.g. the complete bipartite graphK2,4). A sufficient condition that a graph can be drawn convexly is that it is asubdivision of a3-vertex-connected planar graph.Tutte's spring theorem even states that for simple 3-vertex-connected planar graphs the position of the inner vertices can be chosen to be the average of its neighbors.
Word-representable planar graphs include triangle-free planar graphs and, more generally, 3-colourable planar graphs,[10] as well as certain face subdivisions of triangular grid graphs,[11] and certain triangulations of grid-covered cylinder graphs.[12]
Theasymptotic for the number of (labeled) planar graphs on vertices is, where and.[13]
Almost all planar graphs have an exponential number of automorphisms.[14]
The number of unlabeled (non-isomorphic) planar graphs on vertices is between and.[15]
Thefour color theorem states that every planar graph is 4-colorable (i.e., 4-partite).
Fáry's theorem states that every simple planar graph admits a representation as aplanar straight-line graph. Auniversal point set is a set of points such that every planar graph withn vertices has such an embedding with all vertices in the point set; there exist universal point sets of quadratic size, formed by taking a rectangular subset of theinteger lattice. 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, son-vertexregular polygons are universal for outerplanar graphs.
Scheinerman's conjecture (now a theorem) states that every planar graph can be represented as anintersection graph ofline segments in the plane.
Theplanar separator theorem states that everyn-vertex planar graph can be partitioned into twosubgraphs of size at most 2n/3 by the removal of O(√n) vertices. As a consequence, planar graphs also havetreewidth andbranch-width O(√n).
The planar product structure theorem states that every planar graph is a subgraph of the stronggraph product of a graph of treewidth at most 8 and a path.[16]This result has been used to show that planar graphs have boundedqueue number, boundednon-repetitive chromatic number, anduniversal graphs of near-linear size. It also has applications to vertex ranking[17]andp-centered colouring[18]of planar graphs.
For two planar graphs withv vertices, it is possible to determine in time O(v) whether they areisomorphic or not (see alsograph isomorphism problem).[19]
Any planar graph on n nodes has at most 8(n-2) maximal cliques,[20] which implies that the class of planar graphs is aclass with few cliques.
Anapex graph is a graph that may be made planar by the removal of one vertex, and ak-apex graph is a graph that may be made planar by the removal of at mostk vertices.
A1-planar graph is a graph that may be drawn in the plane with at most one simple crossing per edge, and ak-planar graph is a graph that may be drawn with at mostk simple crossings per edge.
Amap graph is a graph formed from a set of finitely many simply-connected interior-disjoint regions in the plane by connecting two regions when they share at least one boundary point. When at most three regions meet at a point, the result is a planar graph, but when four or more regions meet at a point, the result can be nonplanar (for example, if one thinks of a circle divided into sectors, with the sectors being the regions, then the corresponding map graph is the complete graph as all the sectors have a common boundary point - the centre point).
Atoroidal graph is a graph that can be embedded without crossings on thetorus. More generally, thegenus of a graph is the minimum genus of a two-dimensional surface into which the graph may be embedded; planar graphs have genus zero and nonplanar toroidal graphs have genus one. Every graph can be embedded without crossings into some (orientable, connected) closed two-dimensional surface (sphere with handles) and thus the genus of a graph is well defined. Obviously, if the graph can be embedded without crossings into a (orientable, connected, closed) surface with genus g, it can be embedded without crossings into all (orientable, connected, closed) surfaces with greater or equal genus. There are also other concepts in graph theory that are called "X genus" with "X" some qualifier; in general these differ from the above defined concept of "genus" without any qualifier. Especially the non-orientable genus of a graph (using non-orientable surfaces in its definition) is different for a general graph from the genus of that graph (using orientable surfaces in its definition).
Any graph may be embedded into three-dimensional space without crossings. In fact, any graph can be drawn without crossings in a two plane setup, where two planes are placed on top of each other and the edges are allowed to "jump up" and "drop down" from one plane to the other at any place (not just at the graph vertexes) so that the edges can avoid intersections with other edges. This can be interpreted as saying that it is possible to make any electrical conductor network with a two-sidedcircuit board where electrical connection between the sides of the board can be made (as is possible with typical real life circuit boards, with the electrical connections on the top side of the board achieved through pieces of wire and at the bottom side by tracks of copper constructed on to the board itself and electrical connection between the sides of the board achieved through drilling holes, passing the wires through the holes andsoldering them into the tracks); one can also interpret this as saying that in order to build any road network, one only needs just bridges or just tunnels, not both (2 levels is enough, 3 is not needed). Also, in three dimensions the question about drawing the graph without crossings is trivial. However, a three-dimensional analogue of the planar graphs is provided by thelinklessly embeddable graphs, graphs that can be embedded into three-dimensional space in such a way that no two cycles aretopologically linked with each other. In analogy to Kuratowski's and Wagner's characterizations of the planar graphs as being the graphs that do not containK5 orK3,3 as a minor, the linklessly embeddable graphs may be characterized as the graphs that do not contain as a minor any of the seven graphs in thePetersen family. In analogy to the characterizations of the outerplanar and planar graphs as being the graphs withColin de Verdière graph invariant at most two or three, the linklessly embeddable graphs are the graphs that have Colin de Verdière invariant at most four.
Thus a planar graph, when drawn on a flat surface, either has no edge-crossings or can be redrawn without them.