Planar Straight Line Drawing of K5

Graph that tin can be embedded in the airplane

Example graphs
Planar	Nonplanar
Butterfly graph	Complete graph K 5
Consummate graph 1000 4	Utility graph K three,3

In graph theory, a planar graph is a graph that can be embedded in the plane, i.east., it tin can be fatigued 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 manner that no edges cross each other.^{[1] [2]} Such a drawing is called a aeroplane graph or planar embedding of the graph. A plane graph can be defined equally a planar graph with a mapping from every node to a betoken on a plane, and from every edge to a plane 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 tin can exist drawn on the sphere also, and vice versa, past ways of stereographic projection.

Plane graphs can be encoded by combinatorial maps or rotation systems.

An equivalence class of topologically equivalent drawings on the sphere, usually with additional assumptions such as the absence of isthmuses, 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 detail status.

Planar graphs generalize to graphs drawable on a surface of a given genus. In this terminology, planar graphs take genus 0, since the plane (and the sphere) are surfaces of genus 0. Run into "graph embedding" for other related topics.

Planarity criteria [edit]

Kuratowski's and Wagner's theorems [edit]

The Polish mathematician Kazimierz Kuratowski provided a characterization of planar graphs in terms of forbidden graphs, at present known as Kuratowski'southward theorem:

A finite graph is planar if and only if information technology does not contain a subgraph that is a subdivision of the complete graph K _v or the complete bipartite graph ${\displaystyle K_{3,3}}$ (utility graph).

A subdivision of a graph results from inserting vertices into edges (for example, irresolute an border •——• to •—•—•) nix or more times.

An example of a graph with no K ₅ or Thou _three,3 subgraph. Even so, it contains a subdivision of G _3,3 and is therefore non-planar.

Instead of considering subdivisions, Wagner'due south theorem deals with minors:

A finite graph is planar if and just if it does not have ${\displaystyle K_{5}}$ or ${\displaystyle K_{three,3}}$ as a minor.

A minor of a graph results from taking a subgraph and repeatedly contracting an edge into a vertex, with each neighbor of the original end-vertices condign a neighbor of the new vertex.

An animation showing that the Petersen graph contains a minor isomorphic to the K_3,three graph, and is therefore non-planar

Klaus Wagner asked more mostly whether whatsoever 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, ${\displaystyle K_{5}}$ and ${\displaystyle K_{3,three}}$ are the forbidden minors for the class of finite planar graphs.

Other criteria [edit]

In practice, it is hard to utilize Kuratowski'south benchmark to apace decide whether a given graph is planar. However, there exist fast algorithms for this problem: for a graph with northward 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 east edges and f faces, the following simple conditions concur for v ≥ iii:

• Theorem 1. e ≤ 3v − 6;
• Theorem 2. If at that place are no cycles of length iii, and then e ≤ twov − 4.
• Theorem 3. f ≤ 2v − 4.

In this sense, planar graphs are sparse graphs, in that they have but O(v) edges, asymptotically smaller than the maximum O(v ²). The graph G _three,3, for instance, has 6 vertices, 9 edges, and no cycles of length three. Therefore, by Theorem two, it cannot be planar. These theorems provide necessary conditions for planarity that are not sufficient conditions, and therefore tin only be used to bear witness a graph is not planar, non that it is planar. If both theorem 1 and 2 neglect, other methods may be used.

• Whitney'southward planarity criterion gives a characterization based on the existence of an algebraic dual;
• Mac Lane's planarity benchmark gives an algebraic characterization of finite planar graphs, via their cycle spaces;
• The Fraysseix–Rosenstiehl planarity benchmark gives a label based on the existence of a bipartition of the cotree edges of a depth-get-go search tree. Information technology is fundamental 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 benchmark gives a characterization based on the maximum multiplicity of the 2d eigenvalue of certain Schrödinger operators defined by the graph.
• The Hanani–Tutte theorem states that a graph is planar if and but if it has a cartoon in which each independent pair of edges crosses an even number of times; it tin can exist used to characterize the planar graphs via a organisation of equations modulo ii.

Properties [edit]

Euler'due south formula [edit]

Euler's formula states that if a finite, continued, planar graph is drawn in the aeroplane without whatever edge intersections, and five 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), and then

${\displaystyle v-eastward+f=ii.}$

Every bit an illustration, in the butterfly graph given in a higher place, v = 5, e = 6 and f = 3. In general, if the holding holds for all planar graphs of f faces, whatsoever change to the graph that creates an boosted face while keeping the graph planar would keep v −e +f an invariant. Since the belongings holds for all graphs with f = ii, by mathematical induction it holds for all cases. Euler's formula can too exist 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 5 =east + 1 and f = 1, yielding 5 −e +f = 2, i. e., the Euler characteristic is 2.

In a finite, connected, simple, planar graph, whatsoever face (except maybe the outer one) is divisional past at least 3 edges and every edge touches at about 2 faces; using Euler's formula, one can then show that these graphs are sparse in the sense that if v ≥ three:

${\displaystyle e\leq 3v-half dozen.}$

Euler's formula is as well valid for convex polyhedra. This is no coincidence: every convex polyhedron tin can exist turned into a continued, simple, planar graph by using the Schlegel diagram of the polyhedron, a perspective project of the polyhedron onto a plane with the heart of perspective chosen nigh the eye of one of the polyhedron'southward faces. Not every planar graph corresponds to a convex polyhedron in this way: the trees practise not, for example. Steinitz'southward theorem says that the polyhedral graphs formed from convex polyhedra are precisely the finite 3-connected simple planar graphs. More than mostly, Euler's formula applies to whatever polyhedron whose faces are simple polygons that form a surface topologically equivalent to a sphere, regardless of its convexity.

Average degree [edit]

Connected planar graphs with more than one edge obey the inequality ${\displaystyle 2e\geq 3f}$ , because each face has at to the lowest degree three face-border incidences and each edge contributes exactly two incidences. Information technology follows via algebraic transformations of this inequality with Euler'southward formula ${\displaystyle v-e+f=2}$ that for finite planar graphs the average degree is strictly less than 6. Graphs with higher average degree cannot be planar.

Coin graphs [edit]

Example of the circumvolve packing theorem on K⁻ ₅, the consummate graph on five vertices, minus i edge.

We say that ii 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 2 of which accept overlapping interiors, by making a vertex for each circle and an edge for each pair of circles that kiss. The circumvolve packing theorem, first proved by Paul Koebe in 1936, states that a graph is planar if and only if information technology is a money graph.

This result provides an easy proof of Fáry's theorem, that every simple planar graph can exist embedded in the plane in such a fashion that its edges are straight line segments that do not cross each other. If ane places each vertex of the graph at the centre of the corresponding circle in a coin graph representation, then the line segments between centers of kissing circles do non cantankerous whatsoever of the other edges.

Planar graph density [edit]

The meshedness coefficient or density ${\displaystyle D}$ of a planar graph, or network, is the ratio of the number ${\displaystyle f-i}$ of bounded faces (the same every bit the circuit rank of the graph, by Mac Lane's planarity benchmark) by its maximal possible values ${\displaystyle 2v-5}$ for a graph with ${\displaystyle v}$ vertices:

${\displaystyle D={\frac {f-1}{2v-5}}}$

The density obeys ${\displaystyle 0\leq D\leq 1}$ , with ${\displaystyle D=0}$ for a completely sparse planar graph (a tree), and ${\displaystyle D=1}$ for a completely dense (maximal) planar graph.^[iii]

Dual graph [edit]

A planar graph and its dual

Given an embedding Thou of a (non necessarily simple) connected graph in the aeroplane without edge intersections, we construct the dual graph G* as follows: we choose one vertex in each face of M (including the outer face) and for each edge e in 1000 we introduce a new edge in G* connecting the two vertices in Grand* respective to the two faces in G that run into at eastward. Furthermore, this border is drawn so that information technology crosses e exactly once and that no other edge of G or Yard* is intersected. And then K* is over again the embedding of a (not necessarily unproblematic) planar graph; it has as many edges as G, every bit many vertices every bit G has faces and as many faces as G has vertices. The term "dual" is justified past the fact that G** = G; here the equality is the equivalence of embeddings on the sphere. If Yard is the planar graph corresponding to a convex polyhedron, and then One thousand* 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 backdrop of the original graph, enabling results to be proven nigh graphs by examining their dual graphs.

While the dual synthetic for a detail embedding is unique (up to isomorphism), graphs may take different (i.e. not-isomorphic) duals, obtained from unlike (i.east. non-homeomorphic) embeddings.

Families of planar graphs [edit]

Maximal planar graphs [edit]

A unproblematic graph is chosen maximal planar if it is planar just adding any edge (on the given vertex set) would destroy that property. All faces (including the outer one) are and so divisional by three edges, explaining the culling term plane triangulation. The alternative names "triangular graph"^[four] or "triangulated graph"^[5] accept also been used, but are ambiguous, every bit they more usually refer to the line graph of a complete graph and to the chordal graphs respectively. Every maximal planar graph is a to the lowest degree iii-connected.

If a maximal planar graph has 5 vertices with v > 2, then it has precisely 35 − 6 edges and twofive − 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-copse.

Strangulated graphs are the graphs in which every peripheral cycle is a triangle. In a maximal planar graph (or more more often than not a polyhedral graph) the peripheral cycles are the faces, so maximal planar graphs are strangulated. The strangulated graphs include as well the chordal graphs, and are exactly the graphs that tin be formed by clique-sums (without deleting edges) of consummate graphs and maximal planar graphs.^[6]

Outerplanar graphs [edit]

Outerplanar graphs are graphs with an embedding in the plane such that all vertices belong to the unbounded face up of the embedding. Every outerplanar graph is planar, but the converse is non true: Yard ₄ is planar but not outerplanar. A theorem similar to Kuratowski'south states that a finite graph is outerplanar if and only if it does non contain a subdivision of K ₄ or of G _2,three. The to a higher place is a directly corollary of the fact that a graph G is outerplanar if the graph formed from G by adding a new vertex, with edges connecting information technology to all the other vertices, is a planar graph.^[7]

A one-outerplanar embedding of a graph is the aforementioned as an outerplanar embedding. For m > 1 a planar embedding is k-outerplanar if removing the vertices on the outer confront results in a (k − one)-outerplanar embedding. A graph is thousand-outerplanar if it has a g-outerplanar embedding.

Halin graphs [edit]

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 1 confront is side by side to all the others. Every Halin graph is planar. Similar outerplanar graphs, Halin graphs take low treewidth, making many algorithmic bug on them more than easily solved than in unrestricted planar graphs.^[eight]

Upward planar graphs [edit]

An upward planar graph is a directed acyclic graph that tin can be drawn in the plane with its edges as not-crossing curves that are consistently oriented in an upward management. Not every planar directed acyclic graph is upward planar, and it is NP-complete to test whether a given graph is upward planar.

Convex planar graphs [edit]

A planar graph is said to be convex if all of its faces (including the outer confront) are convex polygons. Not all planar graphs have a convex embedding (e.chiliad. the complete bipartite graph ${\displaystyle K_{ii,4}}$ ). A sufficient condition that a graph tin can exist drawn convexly is that it is a subdivision of a 3-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 exist the boilerplate of its neighbors.

Give-and-take-representable planar graphs [edit]

Discussion-representable planar graphs include triangle-costless planar graphs and, more generally, 3-colourable planar graphs,^[nine] also as certain face subdivisions of triangular filigree graphs,^[x] and certain triangulations of grid-covered cylinder graphs.^[11]

Theorems [edit]

Enumeration of planar graphs [edit]

The asymptotic for the number of (labeled) planar graphs on ${\displaystyle n}$ vertices is ${\displaystyle chiliad\cdot n^{-vii/2}\cdot \gamma ^{due north}\cdot n!}$ , where ${\displaystyle \gamma \approx 27.22687}$ and ${\displaystyle g\approx 0.43\times 10^{-v}}$ .^[12]

About all planar graphs take an exponential number of automorphisms.^[13]

The number of unlabeled (not-isomorphic) planar graphs on ${\displaystyle northward}$ vertices is between ${\displaystyle 27.2^{n}}$ and ${\displaystyle thirty.06^{due north}}$ .^[fourteen]

Other results [edit]

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 every bit a planar straight-line graph. A universal point gear up is a set of points such that every planar graph with due north vertices has such an embedding with all vertices in the bespeak 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 stock-still circle and all edges are directly line segments that lie inside the disk and don't intersect, and then n-vertex regular polygons are universal for outerplanar graphs.

Scheinerman's conjecture (at present a theorem) states that every planar graph can be represented every bit an intersection graph of line segments in the plane.

The planar separator theorem states that every n-vertex planar graph can exist partitioned into two subgraphs of size at most 2n/three past the removal of O(√ northward ) vertices. Every bit a consequence, planar graphs likewise take treewidth and branch-width O(√ n ).

The planar product structure theorem states that every planar graph is a subgraph of the potent graph production of a graph of treewidth at near eight and a path.^[xv] This consequence has been used to show that planar graphs have bounded queue number, bounded not-repetitive chromatic number, and universal graphs of most-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 (run across also graph isomorphism trouble).^[xviii]

Generalizations [edit]

An apex graph is a graph that may be made planar past the removal of one vertex, and a k-noon graph is a graph that may be made planar by the removal of at most thou vertices.

A 1-planar graph is a graph that may be drawn in the plane with at most i elementary crossing per edge, and a grand-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 prepare of finitely many simply-connected interior-disjoint regions in the aeroplane by connecting 2 regions when they share at least one boundary indicate. When at most 3 regions run across at a betoken, the result is a planar graph, but when four or more regions meet at a bespeak, the event can be nonplanar.

A toroidal graph is a graph that tin be embedded without crossings on the torus. More more often than not, 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 nil and nonplanar toroidal graphs have genus one.

Any graph may exist embedded into three-dimensional space without crossings. However, a three-dimensional analogue of the planar graphs is provided past the linklessly embeddable graphs, graphs that can be embedded into three-dimensional space in such a way that no two cycles are topologically linked with each other. In analogy to Kuratowski's and Wagner's characterizations of the planar graphs as being the graphs that do non incorporate K _v or K _3,3 equally a small-scale, the linklessly embeddable graphs may be characterized as the graphs that exercise not contain every bit a minor whatever of the seven graphs in the Petersen family. In analogy to the characterizations of the outerplanar and planar graphs as existence the graphs with Colin de Verdière graph invariant at most 2 or three, the linklessly embeddable graphs are the graphs that have Colin de Verdière invariant at about iv.

See also [edit]

• Combinatorial map a combinatorial object that can encode airplane 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 aeroplane
• 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 [edit]

1. ^ Trudeau, Richard J. (1993). Introduction to Graph Theory (Corrected, enlarged republication. ed.). New York: Dover Pub. p. 64. ISBN978-0-486-67870-two . Retrieved 8 August 2012. Thus a planar graph, when drawn on a flat surface, either has no edge-crossings or tin be redrawn without them.
2. ^ Barthelemy, 1000. (2017). Morphogenesis of Spatial Networks. New York: Springer. p. 6.
3. ^ Buhl, J.; Gautrais, J.; Sole, R.V.; Kuntz, P.; Valverde, Southward.; Deneubourg, J.L.; Theraulaz, G. (2004), "Efficiency and robustness in ant networks of galleries", European Physical Periodical B, Springer-Verlag, 42 (i): 123–129, Bibcode:2004EPJB...42..123B, doi:10.1140/epjb/e2004-00364-ix, S2CID 14975826 .
4. ^ Schnyder, Westward. (1989), "Planar graphs and poset dimension", Club, 5 (four): 323–343, doi:x.1007/BF00353652, MR 1010382, S2CID 122785359 .
5. ^ Bhasker, Jayaram; Sahni, Sartaj (1988), "A linear algorithm to detect a rectangular dual of a planar triangulated graph", Algorithmica, three (1–four): 247–278, doi:x.1007/BF01762117, S2CID 2709057 .
6. ^ Seymour, P. D.; Weaver, R. W. (1984), "A generalization of chordal graphs", Journal of Graph Theory, 8 (2): 241–251, doi:10.1002/jgt.3190080206, MR 0742878 .
7. ^ Felsner, Stefan (2004), "one.four Outerplanar Graphs and Convex Geometric Graphs", Geometric graphs and arrangements, Advanced Lectures in Mathematics, Friedr. Vieweg & Sohn, Wiesbaden, pp. 6–vii, doi:ten.1007/978-3-322-80303-0_1, ISBNthree-528-06972-4, MR 2061507
8. ^ Sysło, Maciej K.; Proskurowski, Andrzej (1983), "On Halin graphs", Graph Theory: Proceedings of a Conference held in Lagów, Poland, Feb 10–13, 1981, Lecture Notes in Mathematics, vol. 1018, Springer-Verlag, pp. 248–256, doi:ten.1007/BFb0071635 .
9. ^ "Thou. Halldórsson, S. Kitaev and A. Pyatkin. Semi-transitive orientations and word-representable graphs, Discr. Appl. Math. 201 (2016), 164-171".
10. ^ T. Z. Q. Chen, S. Kitaev, and B. Y. Sun. Word-representability of confront subdivisions of triangular grid graphs, Graphs and Combin. 32(v) (2016), 1749-1761.
11. ^ T. Z. Q. Chen, S. Kitaev, and B. Y. Lord's day. Word-representability of triangulations of grid-covered cylinder graphs, Discr. Appl. Math. 213 (2016), lx-seventy.
12. ^ Giménez, Omer; Noy, Marc (2009). "Asymptotic enumeration and limit laws of planar graphs". Periodical of the American Mathematical Society. 22 (2): 309–329. arXiv:math/0501269. Bibcode:2009JAMS...22..309G. doi:10.1090/s0894-0347-08-00624-iii. S2CID 3353537.
13. ^ McDiarmid, Colin; Steger, Angelika; Welsh, Dominic J.A. (2005). "Random planar graphs". Journal of Combinatorial Theory, Series B. 93 (2): 187–205. CiteSeerX10.1.1.572.857. doi:ten.1016/j.jctb.2004.09.007.
14. ^ Bonichon, N.; Gavoille, C.; Hanusse, N.; Poulalhon, D.; Schaeffer, G. (2006). "Planar Graphs, via Well-Orderly Maps and Trees". Graphs and Combinatorics. 22 (2): 185–202. CiteSeerX10.1.one.106.7456. doi:10.1007/s00373-006-0647-2. S2CID 22639942.
15. ^ Dujmović, Vida; Joret, Gwenäel; Micek, Piotr; Morin, Pat; Ueckerdt, Torsten; Woods, David R. (2020), "Planar graphs have bounded queue number", Periodical of the ACM, 67 (4): 22:1–22:38, arXiv:1904.04791, doi:10.1145/3385731
16. ^ Bose, Prosenjit; Dujmović, Vida; Javarsineh, Mehrnoosh; Morin, Pat (2020), Asymptotically optimal vertex ranking of planar graphs, arXiv:2007.06455
17. ^ Dębski, Michał; Felsner, Stefan; Micek, Piotr; Schröder, Felix (2019), Improved bounds for centered colorings, arXiv:1907.04586
18. ^ I. Southward. Filotti, Jack N. Mayer. A polynomial-time algorithm for determining the isomorphism of graphs of fixed genus. Proceedings of the 12th Annual ACM Symposium on Theory of Computing, p.236–243. 1980.

References [edit]

• Kuratowski, Kazimierz (1930), "Sur le problème des courbes gauches en topologie" (PDF), Fundamenta Mathematicae (in French), 15: 271–283, doi:10.4064/fm-15-one-271-283 .
• Wagner, K. (1937), "Über eine Eigenschaft der ebenen Komplexe", Mathematische Annalen (in High german), 114: 570–590, doi:x.1007/BF01594196, S2CID 123534907 .
• Boyer, John M.; Myrvold, Wendy J. (2005), "On the cut border: Simplified O(due north) planarity by edge addition" (PDF), Periodical of Graph Algorithms and Applications, 8 (iii): 241–273, doi:x.7155/jgaa.00091 .
• McKay, Brendan; Brinkmann, Gunnar, A useful planar graph generator .
• de Fraysseix, H.; Ossona de Mendez, P.; Rosenstiehl, P. (2006), "Trémaux trees and planarity", International Periodical of Foundations of Informatics, 17 (v): 1017–1029, arXiv:math/0610935, doi:10.1142/S0129054106004248, S2CID 40107560 . Special Upshot on Graph Drawing.
• D.A. Bader and S. Sreshta, A New Parallel Algorithm for Planarity Testing, UNM-ECE Technical Report 03-002, Oct 1, 2003.
• Fisk, Steve (1978), "A brusque proof of Chvátal's watchman theorem", Periodical of Combinatorial Theory, Series B, 24 (iii): 374, doi:ten.1016/0095-8956(78)90059-10 .

External links [edit]

• Edge Addition Planarity Algorithm Source Code, version i.0 — Complimentary 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 projection 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 fourth dimension planarity testing, embedding, Kuratowski subgraph isolation, and directly-line drawing.
• 3 Utilities Puzzle and Planar Graphs
• NetLogo Planarity model — NetLogo version of John Tantalo's game

halechimand.blogspot.com

Source: https://en.wikipedia.org/wiki/Planar_graph

Share this post