Number of Planar Drawings of a Complete Graph

Graph that can be embedded in the plane

Example graphs
Planar	Nonplanar
Butterfly graph	Complete graph K 5
Complete graph Thousand iv	Utility graph Thou 3,iii

In graph theory, a planar graph is a graph that can be embedded in the plane, i.e., information technology 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 way that no edges cantankerous each other.^{[1] [2]} Such a drawing is called a plane graph or planar embedding of the graph. A airplane graph can be defined as a planar graph with a mapping from every node to a bespeak on a plane, and from every edge to a plane curve on that plane, such that the extreme points of each bend are the points mapped from its end nodes, and all curves are disjoint except on their farthermost points.

Every graph that can exist drawn on a airplane can be drawn on the sphere as well, and vice versa, by ways of stereographic projection.

Plane graphs tin can exist 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 chosen a planar map. Although a plane graph has an external or unbounded face up, none of the faces of a planar map has a particular status.

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

Planarity criteria [edit]

Kuratowski'southward and Wagner's theorems [edit]

The Polish mathematician Kazimierz Kuratowski provided a characterization of planar graphs in terms of forbidden graphs, now known every bit Kuratowski's theorem:

A finite graph is planar if and but if it does not comprise a subgraph that is a subdivision of the complete graph K _five or the complete bipartite graph ${\displaystyle K_{iii,three}}$ (utility graph).

A subdivision of a graph results from inserting vertices into edges (for example, changing an edge •——• to •—•—•) zero or more than times.

An case of a graph with no One thousand _five or G _3,three subgraph. Even so, information technology contains a subdivision of K _three,3 and is therefore non-planar.

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

A finite graph is planar if and merely if it does non have ${\displaystyle K_{5}}$ or ${\displaystyle K_{3,3}}$ every bit a minor.

A small-scale 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.

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

Klaus Wagner asked more more often than not whether any minor-closed class of graphs is adamant 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, ${\displaystyle K_{v}}$ and ${\displaystyle K_{iii,three}}$ are the forbidden minors for the grade of finite planar graphs.

Other criteria [edit]

In practice, it is difficult to use Kuratowski's benchmark to quickly make up one's mind whether a given graph is planar. However, there be fast algorithms for this trouble: for a graph with n vertices, information technology is possible to make up one's mind in fourth dimension O(n) (linear time) whether the graph may exist planar or non (meet planarity testing).

For a elementary, connected, planar graph with v vertices and e edges and f faces, the following simple conditions hold for five ≥ 3:

• Theorem i. east ≤ 3v − vi;
• Theorem 2. If in that location are no cycles of length 3, then e ≤ 2five − 4.
• Theorem 3. f ≤ 2five − four.

In this sense, planar graphs are sparse graphs, in that they take merely O(v) edges, asymptotically smaller than the maximum O(v ⁱⁱ). The graph Grand _iii,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 atmospheric condition for planarity that are non sufficient conditions, and therefore tin 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.

• Whitney's planarity criterion gives a characterization based on the existence of an algebraic dual;
• Mac Lane'due south planarity criterion gives an algebraic label of finite planar graphs, via their bicycle spaces;
• The Fraysseix–Rosenstiehl planarity criterion gives a label based on the beingness of a bipartition of the cotree edges of a depth-first search tree. Information technology is central to the left-right planarity testing algorithm;
• Schnyder'south 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 tin be used to characterize the planar graphs via a organisation of equations modulo 2.

Properties [edit]

Euler'south formula [edit]

Euler'due south formula states that if a finite, connected, planar graph is drawn in the plane without whatsoever 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), so

${\displaystyle v-e+f=2.}$

Equally an illustration, in the butterfly graph given above, 5 = five, east = six 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 confront while keeping the graph planar would keep five −due east +f an invariant. Since the property holds for all graphs with f = 2, by mathematical induction it holds for all cases. Euler's formula tin can likewise 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 past ane, leaving 5 − e +f constant. Repeat until the remaining graph is a tree; trees have v =e + ane and f = 1, yielding v −e +f = 2, i. e., the Euler characteristic is 2.

In a finite, continued, simple, planar graph, any confront (except maybe the outer one) is divisional by at to the lowest degree 3 edges and every edge touches at most two faces; using Euler'due south formula, ane can and so show that these graphs are thin in the sense that if five ≥ 3:

${\displaystyle e\leq 3v-vi.}$

Euler's formula is also valid for convex polyhedra. This is no coincidence: every convex polyhedron can be turned into a connected, elementary, planar graph past using the Schlegel diagram of the polyhedron, a perspective projection of the polyhedron onto a plane with the eye of perspective chosen near the center of one of the polyhedron'southward faces. Not every planar graph corresponds to a convex polyhedron in this way: the trees do non, for example. Steinitz's theorem says that the polyhedral graphs formed from convex polyhedra are precisely the finite iii-connected elementary planar graphs. More generally, Euler's formula applies to any polyhedron whose faces are simple polygons that form a surface topologically equivalent to a sphere, regardless of its convexity.

Average caste [edit]

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

Coin graphs [edit]

Case of the circle 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 i indicate. A "coin graph" is a graph formed by a ready of circles, no two of which have overlapping interiors, by making a vertex for each circle and an border for each pair of circles that kiss. The circumvolve packing theorem, get-go proved by Paul Koebe in 1936, states that a graph is planar if and merely if it is a coin graph.

This outcome provides an easy proof of Fáry'south theorem, that every simple planar graph can exist embedded in the plane in such a way that its edges are directly line segments that do not cantankerous each other. If i places each vertex of the graph at the middle of the corresponding circumvolve in a coin graph representation, so the line segments between centers of kissing circles do not cross any 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 ${\displaystyle f-1}$ of divisional faces (the same every bit the excursion rank of the graph, by Mac Lane's planarity criterion) by its maximal possible values ${\displaystyle 2v-5}$ for a graph with ${\displaystyle 5}$ 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 ${\displaystyle D=i}$ for a completely dense (maximal) planar graph.^[3]

Dual graph [edit]

A planar graph and its dual

Given an embedding G of a (not necessarily uncomplicated) continued graph in the plane without edge intersections, we construct the dual graph G* as follows: we cull one vertex in each face of G (including the outer confront) and for each border e in Thousand we introduce a new border in G* connecting the two vertices in 1000* corresponding to the two faces in Thousand that meet at east. Furthermore, this edge is drawn so that it crosses east exactly once and that no other edge of Grand or G* is intersected. And 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 past the fact that G** = Yard; here the equality is the equivalence of embeddings on the sphere. If G is the planar graph corresponding to a convex polyhedron, then Grand* is the planar graph corresponding to the dual polyhedron.

Duals are useful because many properties of the dual graph are related in elementary ways to properties of the original graph, enabling results to be proven about graphs by examining their dual graphs.

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

Families of planar graphs [edit]

Maximal planar graphs [edit]

A simple graph is called maximal planar if it is planar but calculation whatsoever edge (on the given vertex prepare) would destroy that property. All faces (including the outer one) are then bounded by 3 edges, explaining the culling term airplane triangulation. The culling names "triangular graph"^[4] or "triangulated graph"^[5] have also been used, just are ambiguous, as they more normally refer to the line graph of a complete graph and to the chordal graphs respectively. Every maximal planar graph is a least 3-connected.

If a maximal planar graph has v vertices with five > 2, then it has precisely iii5 − 6 edges and 2five − 4 faces.

Apollonian networks are the maximal planar graphs formed past repeatedly splitting triangular faces into triples of smaller triangles. Equivalently, they are the planar three-copse.

Strangulated graphs are the graphs in which every peripheral bike 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 likewise the chordal graphs, and are exactly the graphs that can exist formed by clique-sums (without deleting edges) of complete 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 of the embedding. Every outerplanar graph is planar, merely the converse is not true: Chiliad _four 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 K ₄ or of K _two,3. The above is a direct corollary of the fact that a graph G is outerplanar if the graph formed from Thou by adding a new vertex, with edges connecting information technology 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 g > 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 1000-outerplanar if information technology has a k-outerplanar embedding.

Halin graphs [edit]

A Halin graph is a graph formed from an undirected aeroplane tree (with no caste-two nodes) by connecting its leaves into a wheel, in the order given by the plane embedding of the tree. Equivalently, information technology 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 depression treewidth, making many algorithmic bug on them more easily solved than in unrestricted planar graphs.^[viii]

Upwardly planar graphs [edit]

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

Convex planar graphs [edit]

A planar graph is said to exist convex if all of its faces (including the outer face) are convex polygons. Not all planar graphs have a convex embedding (eastward.one thousand. the complete bipartite graph ${\displaystyle K_{2,4}}$ ). A sufficient status that a graph can exist fatigued convexly is that information technology is a subdivision of a three-vertex-continued planar graph. Tutte'due south spring theorem even states that for uncomplicated 3-vertex-connected planar graphs the position of the inner vertices can be chosen to be the boilerplate of its neighbors.

Word-representable planar graphs [edit]

Discussion-representable planar graphs include triangle-costless planar graphs and, more generally, iii-colourable planar graphs,^[9] as well equally certain face subdivisions of triangular grid graphs,^[10] 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 ${\displaystyle yard\cdot n^{-7/ii}\cdot \gamma ^{n}\cdot n!}$ , where ${\displaystyle \gamma \approx 27.22687}$ and ${\displaystyle thou\approx 0.43\times ten^{-5}}$ .^[12]

Almost all planar graphs have an exponential number of automorphisms.^[13]

The number of unlabeled (not-isomorphic) planar graphs on ${\displaystyle n}$ vertices is between ${\displaystyle 27.ii^{due north}}$ and ${\displaystyle xxx.06^{north}}$ .^[14]

Other results [edit]

The four color theorem states that every planar graph is 4-colorable (i.east., iv-partite).

Fáry's theorem states that every simple planar graph admits a representation as a planar directly-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 betoken set up; at that place exist universal indicate 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'southward theorize (now a theorem) states that every planar graph tin be represented equally an intersection graph of line segments in the plane.

The planar separator theorem states that every northward-vertex planar graph tin exist partitioned into two subgraphs of size at most 2n/3 by the removal of O(√ n ) vertices. Every bit a consequence, planar graphs besides have treewidth and branch-width O(√ due north ).

The planar product structure theorem states that every planar graph is a subgraph of the strong graph product of a graph of treewidth at near 8 and a path.^[15] This effect has been used to prove that planar graphs accept bounded queue number, bounded non-repetitive chromatic number, and universal graphs of virtually-linear size. It too has applications to vertex ranking^[xvi] and p-centered colouring^[17] of planar graphs.

For two planar graphs with v vertices, it is possible to determine in time O(5) whether they are isomorphic or non (run across also graph isomorphism problem).^[xviii]

Generalizations [edit]

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

A ane-planar graph is a graph that may be drawn in the plane with at about one unproblematic crossing per edge, and a k-planar graph is a graph that may exist drawn with at most k simple crossings per edge.

A map graph is a graph formed from a fix of finitely many simply-continued interior-disjoint regions in the aeroplane by connecting two regions when they share at least one boundary point. When at virtually three regions meet at a point, the result is a planar graph, but when 4 or more regions meet at a indicate, the result can be nonplanar.

A toroidal graph is a graph that tin can be embedded without crossings on the torus. More generally, the genus of a graph is the minimum genus of a ii-dimensional surface into which the graph may exist embedded; planar graphs have genus zero and nonplanar toroidal graphs have genus ane.

Any graph may be embedded into three-dimensional infinite without crossings. However, a three-dimensional analogue of the planar graphs is provided past the linklessly embeddable graphs, graphs that tin be embedded into three-dimensional space in such a way that no 2 cycles are topologically linked with each other. In analogy to Kuratowski's and Wagner's characterizations of the planar graphs equally beingness the graphs that exercise not incorporate K _five or One thousand _iii,3 as a pocket-sized, the linklessly embeddable graphs may be characterized as the graphs that practice not comprise as a small-scale whatever of the seven graphs in the Petersen family. In analogy to the characterizations of the outerplanar and planar graphs equally being the graphs with Colin de Verdière graph invariant at most ii or three, the linklessly embeddable graphs are the graphs that have Colin de Verdière invariant at most iv.

See also [edit]

• Combinatorial map a combinatorial object that can encode plane graphs
• Planarization, a planar graph formed from a drawing with crossings past 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 synthetic as part of the game play
• 3 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-ii . Retrieved viii August 2012. Thus a planar graph, when drawn on a apartment surface, either has no edge-crossings or can be redrawn without them.
2. ^ Barthelemy, M. (2017). Morphogenesis of Spatial Networks. New York: Springer. p. 6.
3. ^ Buhl, J.; Gautrais, J.; Sole, R.Five.; Kuntz, P.; Valverde, S.; Deneubourg, J.L.; Theraulaz, One thousand. (2004), "Efficiency and robustness in ant networks of galleries", European Physical Journal B, Springer-Verlag, 42 (1): 123–129, Bibcode:2004EPJB...42..123B, doi:x.1140/epjb/e2004-00364-nine, S2CID 14975826 .
4. ^ Schnyder, Westward. (1989), "Planar graphs and poset dimension", Order, 5 (four): 323–343, doi:10.1007/BF00353652, MR 1010382, S2CID 122785359 .
5. ^ Bhasker, Jayaram; Sahni, Sartaj (1988), "A linear algorithm to find a rectangular dual of a planar triangulated graph", Algorithmica, 3 (1–iv): 247–278, doi:x.1007/BF01762117, S2CID 2709057 .
6. ^ Seymour, P. D.; Weaver, R. Due west. (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), "1.iv Outerplanar Graphs and Convex Geometric Graphs", Geometric graphs and arrangements, Advanced Lectures in Mathematics, Friedr. Vieweg & Sohn, Wiesbaden, pp. half-dozen–7, doi:10.1007/978-3-322-80303-0_1, ISBN3-528-06972-4, MR 2061507
8. ^ Sysło, Maciej M.; Proskurowski, Andrzej (1983), "On Halin graphs", Graph Theory: Proceedings of a Conference held in Lagów, Poland, February 10–xiii, 1981, Lecture Notes in Mathematics, vol. 1018, Springer-Verlag, pp. 248–256, doi:10.1007/BFb0071635 .
9. ^ "M. 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. Sunday. Give-and-take-representability of face subdivisions of triangular grid graphs, Graphs and Combin. 32(5) (2016), 1749-1761.
11. ^ T. Z. Q. Chen, S. Kitaev, and B. Y. Sun. Word-representability of triangulations of filigree-covered cylinder graphs, Discr. Appl. Math. 213 (2016), 60-70.
12. ^ Giménez, Omer; Noy, Marc (2009). "Asymptotic enumeration and limit laws of planar graphs". Periodical of the American Mathematical Society. 22 (two): 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.one.1.572.857. doi:10.1016/j.jctb.2004.09.007.
14. ^ Bonichon, Due north.; Gavoille, C.; Hanusse, North.; Poulalhon, D.; Schaeffer, G. (2006). "Planar Graphs, via Well-Orderly Maps and Trees". Graphs and Combinatorics. 22 (2): 185–202. CiteSeerX10.1.1.106.7456. doi:x.1007/s00373-006-0647-two. S2CID 22639942.
15. ^ Dujmović, Vida; Joret, Gwenäel; Micek, Piotr; Morin, Pat; Ueckerdt, Torsten; Wood, David R. (2020), "Planar graphs have bounded queue number", Journal 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. South. Filotti, Jack North. Mayer. A polynomial-time algorithm for determining the isomorphism of graphs of fixed genus. Proceedings of the 12th Annual ACM Symposium on Theory of Calculating, 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-15-1-271-283 .
• Wagner, One thousand. (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 cutting edge: Simplified O(northward) planarity past edge addition" (PDF), Journal of Graph Algorithms and Applications, eight (3): 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 copse and planarity", International Periodical of Foundations of Information science, 17 (5): 1017–1029, arXiv:math/0610935, doi:10.1142/S0129054106004248, S2CID 40107560 . Special Issue on Graph Drawing.
• D.A. Bader and Due south. Sreshta, A New Parallel Algorithm for Planarity Testing, UNM-ECE Technical Study 03-002, Oct 1, 2003.
• Fisk, Steve (1978), "A short proof of Chvátal'southward watchman theorem", Journal of Combinatorial Theory, Series B, 24 (3): 374, doi:10.1016/0095-8956(78)90059-X .

External links [edit]

• Edge Addition Planarity Algorithm Source Lawmaking, version i.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 Add-on Planarity Algorithms, electric 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 direct-line drawing.
• three Utilities Puzzle and Planar Graphs
• NetLogo Planarity model — NetLogo version of John Tantalo'south game

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

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