Planar graph
Example graphs | |
---|---|
Planar | Nonplanar |
Butterfly graph |
Complete graph K5 |
Complete graph K4 |
Utility graph K3,3
|
In
Every graph that can be drawn on a plane can be drawn on the sphere as well, and vice versa, by means of stereographic projection.
Plane graphs can be encoded by combinatorial maps or rotation systems.
An
Planar graphs generalize to graphs drawable on a surface of a given
Planarity criteria
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 utility graph).
A
Instead of considering subdivisions,
- A finite graph is planar if and only if it does not have K5 or K3,3 as a minor.
A
Other 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).
For a simple, connected, planar graph with v vertices and e edges and f faces, the following simple conditions hold for v ≥ 3:
- Theorem 1. e ≤ 3v – 6;
- Theorem 2. If there are no cycles of length 3, then e ≤ 2v – 4.
- Theorem 3. f ≤ 2v – 4.
In this sense, planar graphs are
- Whitney's planarity criterion gives a characterization based on the existence of an algebraic dual;
- Mac Lane's planarity criterion gives an algebraic characterization of finite planar graphs, via their cycle spaces;
- The Fraysseix–Rosenstiehl 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 testingalgorithm;
- 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.
- The Hanani–Tutte theorem states that a graph is planar if and only if it has a drawing in which each independent pair of edges crosses an even number of times; it can be used to characterize the planar graphs via a system of equations modulo 2.
Properties
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
As an illustration, in the butterfly graph given above, v = 5, e = 6 and f = 3. 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 proved 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,
Euler's formula is also valid for
Average degree
Connected planar graphs with more than one edge obey the inequality 2e ≥ 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 formula v – e + f = 2 that for finite planar graphs the average degree is strictly less than 6. Graphs with higher average degree cannot be planar.
Coin graphs
We say that two circles drawn in a plane kiss (or osculate) 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. The circle packing theorem, first proved by Paul Koebe in 1936, states that a graph is planar if and only if it is a coin graph.
This result provides an easy proof of Fáry's theorem, that every simple planar graph can be embedded in the plane in such a way that its edges are straight line 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.
Planar graph density
The meshedness coefficient or density D of a planar graph, or network, is the ratio of the number f – 1 of bounded faces (the same as the circuit rank of the graph, by Mac Lane's planarity criterion) by its maximal possible values 2v – 5 for a graph with v vertices:
The density obeys 0 ≤ D ≤ 1, with D = 0 for a completely sparse planar graph (a tree), and D = 1 for a completely dense (maximal) planar graph.[3]
Dual graph
Given an embedding G of a (not necessarily simple) connected 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.
While the dual constructed for a particular embedding is unique (up to
Families of planar graphs
Maximal planar graphs
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 (including the outer one) are then bounded by three edges, explaining the alternative term plane triangulation. The alternative names "triangular graph"[4] or "triangulated graph"[5] have also been used, but are ambiguous, as they more commonly refer to the line graph of a complete graph and to the chordal graphs respectively. Every maximal planar graph is at least 3-connected.
If a maximal planar graph has v vertices with v > 2, then it has precisely 3v – 6 edges and 2v – 4 faces.
Apollonian networks are the maximal planar graphs formed by repeatedly splitting triangular faces into triples of smaller triangles. Equivalently, they are the planar 3-trees.
Strangulated graphs are the graphs in which every peripheral 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 the chordal graphs, and are exactly the graphs that can be formed by clique-sums (without deleting edges) of complete graphs and maximal planar graphs.[6]
Outerplanar 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. The above is a direct corollary of the fact that a graph G is outerplanar if the graph formed from G by adding a new vertex, with edges connecting it to all the other vertices, is a planar graph.[7]
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.
Halin graphs
A Halin 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 a polyhedral graph in which one face is adjacent to all the others. Every Halin graph is planar. Like outerplanar graphs, Halin graphs have low treewidth, making many algorithmic problems on them more easily solved than in unrestricted planar graphs.[8]
Upward planar graphs
An
Convex planar graphs
A planar graph is said to be convex if all of its faces (including the outer face) are
Word-representable planar graphs
Word-representable planar graphs include triangle-free planar graphs and, more generally, 3-colourable planar graphs,[9] as well as certain face subdivisions of triangular grid graphs,[10] and certain triangulations of grid-covered cylinder graphs.[11]
Theorems
Enumeration of planar graphs
The asymptotic for the number of (labeled) planar graphs on vertices is , where and .[12]
Almost all planar graphs have an exponential number of automorphisms.[13]
The number of unlabeled (non-isomorphic) planar graphs on vertices is between and .[14]
Other results
The four 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 a planar straight-line graph. A universal point set is a set of points such that every planar graph with n 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 the integer 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, so n-vertex regular polygons are universal for outerplanar graphs.
Scheinerman's conjecture (now a theorem) states that every planar graph can be represented as an intersection graph of line segments in the plane.
The
The planar product structure theorem states that every planar graph is a subgraph of the strong graph product of a graph of treewidth at most 8 and a path.[15] This result has been used to show that planar graphs have bounded queue number, bounded non-repetitive chromatic number, and universal graphs of near-linear size. It also has applications to vertex ranking[16] and p-centered colouring[17] of planar graphs.
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).[18]
Any planar graph on n nodes has at most 8(n-2) maximal cliques,[19] which implies that the class of planar graphs is a class with few cliques.
Generalizations
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.
A 1-planar graph is a graph that may be drawn in the plane with at most one simple crossing per edge, and a k-planar graph is a graph that may be drawn with at most k simple crossings per edge.
A map 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).
A toroidal graph is a graph that can be embedded without crossings on the torus. More generally, the genus 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-sided
See also
- Combinatorial map a combinatorial object that can encode plane graphs
- Planarization, a planar graph formed from a drawing with crossings by replacing each crossing point by a new vertex
- Thickness (graph theory), the smallest number of planar graphs into which the edges of a given graph may be partitioned
- Planarity, a puzzle computer game in which the objective is to embed a planar graph onto a plane
- Sprouts (game), a pencil-and-paper game where a planar graph subject to certain constraints is constructed as part of the game play
- Three utilities problem, a popular puzzle
Notes
- ISBN 978-0-486-67870-2. Retrieved 8 August 2012.
Thus a planar graph, when drawn on a flat surface, either has no edge-crossings or can be redrawn without them.
- ISBN 978-3-319-20565-6.
- S2CID 14975826.
- S2CID 122785359.
- S2CID 2709057.
- MR 0742878.
- MR 2061507
- .
- S2CID 26796091.
- S2CID 43817300.
- S2CID 26987743.
- S2CID 3353537.
- .
- S2CID 22639942.
- doi:10.1145/3385731
- arXiv:2007.06455
- S2CID 195874032
- S2CID 16345164.
- ^ Wood, D. R. (2007). On the Maximum Number of Cliques in a Graph. Graphs and Combinatorics, 23(3), 337–352. https://doi.org/10.1007/s00373-007-0738-8
References
- .
- Wagner, K. (1937), "Über eine Eigenschaft der ebenen Komplexe", Mathematische Annalen (in German), 114: 570–590, S2CID 123534907.
- Boyer, John M.; .
- McKay, Brendan; Brinkmann, Gunnar, A useful planar graph generator.
- de Fraysseix, H.; S2CID 40107560. Special Issue on Graph Drawing.
- Bader, D.A.; Sreshta, S. (October 1, 2003). A New Parallel Algorithm for Planarity Testing (Technical report). UNM-ECE Technical Report 03-002. Archived from the original on 2016-03-16.
- Fisk, Steve (1978), "A short proof of Chvátal's watchman theorem", Journal of Combinatorial Theory, Series B, 24 (3): 374, .
External links
- Edge Addition Planarity Algorithm Source Code, version 1.0 — Free C source code for reference implementation of Boyer–Myrvold planarity algorithm, which provides both a combinatorial planar embedder and Kuratowski subgraph isolator. An open source project with free licensing provides the Edge Addition Planarity Algorithms, current version.
- 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.
- Boost Graph Library tools for planar graphs, including linear time planarity testing, embedding, Kuratowski subgraph isolation, and straight-line drawing.
- 3 Utilities Puzzle and Planar Graphs
- NetLogo Planarity model — NetLogo version of John Tantalo's game