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graph theory In mathematics, graph theory is the study of '' graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of '' vertices'' (also called ''nodes'' or ''points'') which are conn ...
, a planar graph is a graph that can be 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 words, it can be drawn in such a way that no edges cross each other. Such a drawing is called a plane graph or planar 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 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 can be drawn on the
sphere A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the c ...
as well, and vice versa, by means of
stereographic projection In mathematics, a stereographic projection is a perspective projection of the sphere, through a specific point on the sphere (the ''pole'' or ''center of projection''), onto a plane (the ''projection plane'') perpendicular to the diameter thro ...
. Plane graphs can be encoded by
combinatorial map A combinatorial map is a combinatorial representation of a graph on an orientable surface. A combinatorial map may also be called a combinatorial embedding, a rotation system, an orientable ribbon graph, a fat graph, or a cyclic graph. More general ...
s or
rotation system In combinatorial mathematics, rotation systems (also called combinatorial embeddings or combinatorial maps) encode embeddings of graphs onto orientable surfaces by describing the circular ordering of a graph's edges around each vertex. A more for ...
s. An
equivalence class In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements a ...
of
topologically equivalent In mathematics, two functions are said to be topologically conjugate if there exists a homeomorphism that will conjugate the one into the other. Topological conjugacy, and related-but-distinct of flows, are important in the study of iterated fu ...
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 particular status. Planar graphs generalize to graphs drawable on a surface of a given
genus Genus ( plural genera ) is a taxonomic rank used in the biological classification of living and fossil organisms as well as viruses. In the hierarchy of biological classification, genus comes above species and below family. In binomial nom ...
. In this terminology, planar graphs have
genus Genus ( plural genera ) is a taxonomic rank used in the biological classification of living and fossil organisms as well as viruses. In the hierarchy of biological classification, genus comes above species and below family. In binomial nom ...
 0, since the plane (and the sphere) are surfaces of genus 0. See " graph embedding" for other related topics.


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 In graph theory, Kuratowski's theorem is a mathematical forbidden graph characterization of planar graphs, named after Kazimierz Kuratowski. It states that a finite graph is planar if and only if it does not contain a subgraph that is a subd ...
: :A finite graph is planar
if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bic ...
it does not contain a subgraph that is a subdivision of the complete graph or the complete bipartite graph ( utility graph). A subdivision of a graph results from inserting vertices into edges (for example, changing an edge zero or more times. Instead of considering subdivisions,
Wagner's theorem In graph theory, Wagner's theorem is a mathematical forbidden graph characterization of planar graphs, named after Klaus Wagner, stating that a finite graph is planar if and only if its minors include neither ''K''5 (the complete graph on five ...
deals with minors: :A finite graph is planar if and only if it does not have or 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 becoming a neighbor of the new vertex.
Klaus Wagner Klaus Wagner (March 31, 1910 – February 6, 2000) was a German mathematician known for his contributions to graph theory. Education and career Wagner studied topology at the University of Cologne under the supervision of who had been a stude ...
asked more generally whether any minor-closed class of graphs is determined by a finite set of "
forbidden minor In graph theory, a branch of mathematics, many important families of graphs can be described by a finite set of individual graphs that do not belong to the family and further exclude all graphs from the family which contain any of these forbidde ...
s". This is now the Robertson–Seymour theorem, proved in a long series of papers. In the language of this theorem, and are the forbidden minors for the class of finite planar graphs.


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
algorithm In mathematics and computer science, an algorithm () is a finite sequence of rigorous instructions, typically used to solve a class of specific problems or to perform a computation. Algorithms are used as specifications for performing ...
s for this problem: for a graph with vertices, it is possible to determine in time (linear time) whether the graph may be planar or not (see planarity testing). For a simple, connected, planar graph with vertices and edges and faces, the following simple conditions hold for : * Theorem 1. ; * Theorem 2. If there are no cycles of length 3, then . * Theorem 3. . In this sense, planar graphs are sparse graphs, in that they have only edges, asymptotically smaller than the maximum . The graph , 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. * 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 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. * 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 is the number of vertices, is the number of edges and 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 In the mathematical field of graph theory, the butterfly graph (also called the bowtie graph and the hourglass graph) is a planar, undirected graph with 5 vertices and 6 edges. It can be constructed by joining 2 copies of the cycle graph with ...
given above, , and . In general, if the property holds for all planar graphs of faces, any change to the graph that creates an additional face while keeping the graph planar would keep an invariant. Since the property holds for all graphs with , by
mathematical induction Mathematical induction is a method for proving that a statement ''P''(''n'') is true for every natural number ''n'', that is, that the infinitely many cases ''P''(0), ''P''(1), ''P''(2), ''P''(3), ...  all hold. Informal metaphors help ...
it holds for all cases. Euler's formula can also be proved as follows: if the graph isn't a
tree In botany, a tree is a perennial plant with an elongated stem, or trunk, usually supporting branches and leaves. In some usages, the definition of a tree may be narrower, including only woody plants with secondary growth, plants that are ...
, then remove an edge which completes a cycle. This lowers both and by one, leaving constant. Repeat until the remaining graph is a tree; trees have and , yielding , i. e., the
Euler characteristic In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological spac ...
is 2. In a finite, connected, ''
simple Simple or SIMPLE may refer to: *Simplicity, the state or quality of being simple Arts and entertainment * ''Simple'' (album), by Andy Yorke, 2008, and its title track * "Simple" (Florida Georgia Line song), 2018 * "Simple", a song by Johnn ...
'', 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 if : :e\leq 3v-6. Euler's formula is also valid for convex polyhedra. This is no coincidence: every convex polyhedron can be turned into a connected, simple, planar graph by using the Schlegel diagram of the polyhedron, a
perspective projection Linear or point-projection perspective (from la, perspicere 'to see through') is one of two types of graphical projection perspective in the graphic arts; the other is parallel projection. Linear perspective is an approximate representation ...
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 In polyhedral combinatorics, a branch of mathematics, Steinitz's theorem is a characterization of the undirected graphs formed by the edges and vertices of three-dimensional convex polyhedra: they are exactly the 3-vertex-connected planar gra ...
says that the polyhedral graphs formed from convex polyhedra are precisely the finite 3-connected simple planar graphs. More generally, Euler's formula applies to any polyhedron whose faces are simple polygons that form a surface
topologically equivalent In mathematics, two functions are said to be topologically conjugate if there exists a homeomorphism that will conjugate the one into the other. Topological conjugacy, and related-but-distinct of flows, are important in the study of iterated fu ...
to a sphere, regardless of its convexity.


Average degree

Connected planar graphs with more than one edge obey the inequality , 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 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 In graph theory, the meshedness coefficient is a graph invariant of planar graphs that measures the number of bounded faces of the graph, as a fraction of the possible number of faces for other planar graphs with the same number of vertices. It rang ...
or density of a planar graph, or network, is the ratio of the number of bounded faces (the same as the circuit rank of the graph, by Mac Lane's planarity criterion) by its maximal possible values for a graph with vertices: :D = \frac The density obeys , with for a completely sparse planar graph (a tree), and for a completely dense (maximal) planar graph.


Dual graph

Given an embedding of a (not necessarily simple) connected graph in the plane without edge intersections, we construct the '' dual graph'' as follows: we choose one vertex in each face of (including the outer face) and for each edge in we introduce a new edge in connecting the two vertices in corresponding to the two faces in that meet at . Furthermore, this edge is drawn so that it crosses exactly once and that no other edge of or is intersected. Then is again the embedding of a (not necessarily simple) planar graph; it has as many edges as , as many vertices as has faces and as many faces as has vertices. The term "dual" is justified by the fact that ; here the equality is the equivalence of embeddings on the
sphere A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the c ...
. If is the planar graph corresponding to a convex polyhedron, then 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
isomorphism In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
), graphs may have different (i.e. non-isomorphic) duals, obtained from different (i.e. non-
homeomorphic In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomor ...
) embeddings.


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" or "triangulated graph" have also been used, but are ambiguous, as they more commonly refer to the line graph of a complete graph and to the
chordal graph In the mathematical area of graph theory, a chordal graph is one in which all cycles of four or more vertices have a ''chord'', which is an edge that is not part of the cycle but connects two vertices of the cycle. Equivalently, every induced ...
s respectively. Every maximal planar graph is a least 3-connected. If a maximal planar graph has vertices with , then it has precisely edges and faces.
Apollonian network In combinatorial mathematics, an Apollonian network is an undirected graph formed by a process of recursively subdividing a triangle into three smaller triangles. Apollonian networks may equivalently be defined as the planar 3-trees, the maxima ...
s 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 In graph theory, a peripheral cycle (or peripheral circuit) in an undirected graph is, intuitively, a cycle that does not separate any part of the graph from any other part. Peripheral cycles (or, as they were initially called, peripheral polyg ...
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 graph In the mathematical area of graph theory, a chordal graph is one in which all cycles of four or more vertices have a ''chord'', which is an edge that is not part of the cycle but connects two vertices of the cycle. Equivalently, every induced ...
s, and are exactly the graphs that can be formed by clique-sums (without deleting edges) of complete graphs and maximal planar graphs.


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: 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 or of . The above is a direct corollary of the fact that a graph is outerplanar if the graph formed from by adding a new vertex, with edges connecting it to all the other vertices, is a planar graph. A 1-outerplanar embedding of a graph is the same as an outerplanar embedding. For a planar embedding is -outerplanar if removing the vertices on the outer face results in a -outerplanar embedding. A graph is -outerplanar if it has a -outerplanar embedding.


Halin graphs

A
Halin graph In graph theory, a Halin graph is a type of planar graph, constructed by connecting the leaves of a tree into a cycle. The tree must have at least four vertices, none of which has exactly two neighbors; it should be drawn in the plane so none of ...
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.


Upward planar graphs

An upward planar graph is a
directed acyclic graph In mathematics, particularly graph theory, and computer science, a directed acyclic graph (DAG) is a directed graph with no directed cycles. That is, it consists of vertices and edges (also called ''arcs''), with each edge directed from one ...
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 is NP-complete to test whether a given graph is upward planar.


Convex planar graphs

A planar graph is said to be convex if all of its faces (including the outer face) are convex polygons. Not all planar graphs have a convex embedding (e.g. the complete bipartite graph ). A sufficient condition that a graph can be 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 be the average of its neighbors.


Word-representable planar graphs

Word-representable planar graphs include triangle-free planar graphs and, more generally, 3-colourable planar graphs, as well as certain face subdivisions of triangular grid graphs, and certain triangulations of grid-covered cylinder graphs.


Theorems


Enumeration of planar graphs

The
asymptotic In analytic geometry, an asymptote () of a curve is a line such that the distance between the curve and the line approaches zero as one or both of the ''x'' or ''y'' coordinates tends to infinity. In projective geometry and related context ...
for the number of (labeled) planar graphs on n vertices is g\cdot n^\cdot \gamma^n\cdot n!, where \gamma\approx 27.22687 and g\approx 0.43\times 10^. Almost all planar graphs have an exponential number of automorphisms. The number of unlabeled (non-isomorphic) planar graphs on n vertices is between 27.2^n and 30.06^n.


Other results

The
four color theorem In mathematics, the four color theorem, or the four color map theorem, states that no more than four colors are required to color the regions of any map so that no two adjacent regions have the same color. ''Adjacent'' means that two regions sha ...
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 In computational geometry and geometric graph theory, a planar straight-line graph, in short ''PSLG'', (or ''straight-line plane graph'', or ''plane straight-line graph'') is a term used for an embedding of a planar graph in the plane such that ...
. A
universal point set In graph drawing, a universal point set of order ''n'' is a set ''S'' of points in the Euclidean plane with the property that every ''n''-vertex planar graph has a straight-line drawing in which the vertices are all placed at points of ''S''. Bo ...
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 In mathematics, Scheinerman's conjecture, now a theorem, states that every planar graph is the intersection graph of a set of line segments in the plane. This conjecture was formulated by E. R. Scheinerman in his Ph.D. thesis (1984), following e ...
(now a theorem) states that every planar graph can be represented as an intersection graph of line segments in the plane. The planar separator theorem states that every ''n''-vertex planar graph can be partitioned into two subgraphs of size at most 2''n''/3 by the removal of O() vertices. As a consequence, planar graphs also have treewidth and
branch-width In graph theory, a branch-decomposition of an undirected graph ''G'' is a hierarchical clustering of the edges of ''G'', represented by an unrooted binary tree ''T'' with the edges of ''G'' as its leaves. Removing any edge from ''T'' partitions t ...
O(). The planar product structure theorem states that every planar graph is a subgraph of the strong
graph product In graph theory, a graph product is a binary operation on graphs. Specifically, it is an operation that takes two graphs and and produces a graph with the following properties: * The vertex set of is the Cartesian product , where and are ...
of a graph of treewidth at most 8 and a path. This result has been used to show that planar graphs have bounded queue number, bounded non-repetitive chromatic number, and
universal graph In mathematics, a universal graph is an infinite graph that contains ''every'' finite (or at-most-countable) graph as an induced subgraph. A universal graph of this type was first constructed by Richard Rado and is now called the Rado graph or ...
s of near-linear size. It also has applications to vertex ranking and ''p''-centered colouring of planar graphs. For two planar graphs with ''v'' vertices, it is possible to determine in time O(''v'') whether they are
isomorphic In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
or not (see also graph isomorphism problem).


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. A toroidal graph is a graph that can be embedded without crossings on the
torus In geometry, a torus (plural tori, colloquially donut or doughnut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle. If the axis of revolution does n ...
. More generally, the
genus Genus ( plural genera ) is a taxonomic rank used in the biological classification of living and fossil organisms as well as viruses. In the hierarchy of biological classification, genus comes above species and below family. In binomial nom ...
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. Any graph may be embedded into three-dimensional space without crossings. However, a three-dimensional analogue of the planar graphs is provided by 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 not contain ''K''5 or ''K''3,3 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 the
Petersen family In graph theory, the Petersen family is a set of seven undirected graphs that includes the Petersen graph and the complete graph . The Petersen family is named after Danish mathematician Julius Petersen, the namesake of the Petersen graph. Any o ...
. In analogy to the characterizations of the outerplanar and planar graphs as being the graphs with Colin 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.


See also

*
Combinatorial map A combinatorial map is a combinatorial representation of a graph on an orientable surface. A combinatorial map may also be called a combinatorial embedding, a rotation system, an orientable ribbon graph, a fat graph, or a cyclic graph. More general ...
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) Sprouts is a paper-and-pencil game which can be analyzed for its mathematical properties. It was invented by mathematicians John Horton Conway and Michael S. Paterson at Cambridge University in the early 1960s. The setup is even simpler than the ...
, 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


References

*. *. *. *. *. Special Issue on Graph Drawing. * *.


External links

{{commons category, Planar graphs
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 th
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.

including linear time planarity testing, embedding, Kuratowski subgraph isolation, and straight-line drawing.
3 Utilities Puzzle and Planar GraphsNetLogo Planarity model
— NetLogo version of John Tantalo's game Graph families Intersection classes of graphs