Example graphs | |
---|---|

Planar | Nonplanar |

Butterfly graph | Complete graph K_{5} |

Complete graph K_{4} | Utility graph K_{3,3} |

In graph theory, 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.^{ [1] }^{ [2] } 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.

- Kuratowski's and Wagner's theorems
- Other planarity criteria
- Euler's formula
- Average degree
- Coin graphs
- Planar graph density
- Related families of graphs
- Maximal planar graphs
- Outerplanar graphs
- Halin graphs
- Other related families
- Enumeration of planar graphs
- Other facts and definitions
- See also
- Notes
- References
- External links

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 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 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 plane (and the sphere) are surfaces of genus 0. See "graph embedding" for other related topics.

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

- A finite graph is planar if and only if it does not contain a subgraph that is a subdivision of the complete graph
*K*_{5}or the complete bipartite graph (utility graph).

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

Instead of considering subdivisions, Wagner's theorem deals 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 asked more generally whether any 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, and are the forbidden minors for the class of finite planar graphs.

In practice, it is difficult to use Kuratowski's criterion to quickly decide whether a given graph is planar. However, there exist 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*≤ 3*v*− 6; - Theorem 2. If there are no cycles of length 3, then
*e*≤ 2*v*− 4. - Theorem 3.
*f*≤ 2*v*− 4.

In this sense, planar graphs are sparse graphs, in that they have only O(*v*) edges, asymptotically smaller than the maximum O(*v*^{2}). The graph *K*_{3,3}, for example, has 6 vertices, 9 edges, and no cycles of length 3. Therefore, by Theorem 2, it cannot be planar. These theorems provide necessary conditions for planarity that are not sufficient conditions, and therefore can only be used to prove a graph is not planar, not that it is planar. If both theorem 1 and 2 fail, other methods may be used.

- 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.

**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, connected, * simple *, planar graph, any face (except possibly the outer one) is bounded by at least three edges and every edge touches at most two faces; using Euler's formula, one can then show that these graphs are *sparse* in the sense that if *v* ≥ 3:

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 of the polyhedron onto a plane with the center of perspective chosen near the center of one of the polyhedron's faces. Not every planar graph corresponds to a convex polyhedron in this way: the trees do not, for example. Steinitz's theorem says that 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 to a sphere, regardless of its convexity.

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.

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.

The density of a planar graph, or network, is defined as a ratio of the number of edges to the number of possible edges in a network with nodes, given by a planar graph , giving . A completely sparse planar graph has , alternatively a completely dense planar graph has

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"^{ [3] } or "triangulated graph"^{ [4] } 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 a least 3-connected.

If a maximal planar graph has *v* vertices with *v* > 2, then it has precisely 3*v* − 6 edges and 2*v* − 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.^{ [5] }

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: *K*_{4} 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*_{2,3}. 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.^{ [6] }

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.

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.^{ [7] }

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. 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.

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 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.

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

The asymptotic for the number of (labeled) planar graphs on vertices is , where and .^{ [8] }

Almost all planar graphs have an exponential number of automorphisms.^{ [9] }

The number of unlabeled (non-isomorphic) planar graphs on vertices is between and .^{ [10] }

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 an embedding in the plane such that all edges are straight line segments which don't intersect. 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.

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 isomorphism), graphs may have different (i.e. non-isomorphic) duals, obtained from different (i.e. non-homeomorphic) embeddings.

A *Euclidean graph* is a graph in which the vertices represent points in the plane, and the edges are assigned lengths equal to the Euclidean distance between those points; see Geometric graph theory.

A plane graph is said to be *convex* if all of its faces (including the outer face) are convex polygons. A planar graph may be drawn convexly if and only if it is a subdivision of a 3-vertex-connected planar graph.

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 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(√*n*) vertices. As a consequence, planar graphs also have treewidth and branch-width O(√*n*).

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.^{ [11] } 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^{ [12] } and $p$-centered colouring^{ [13] } 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).^{ [14] }

The meshedness coefficient of a planar graph normalizes its number of bounded faces (the same as the circuit rank of the graph, by Mac Lane's planarity criterion) by dividing it by 2*n* − 5, the maximum possible number of bounded faces in a planar graph with *n* vertices. Thus, it ranges from 0 for trees to 1 for maximal planar graphs.^{ [15] }

Word-representable planar graphs include triangle-free planar graphs and, more generally, 3-colourable planar graphs,^{ [16] } as well as certain face subdivisions of triangular grid graphs,^{ [17] } and certain triangulations of grid-covered cylinder graphs.^{ [18] }

- 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

- ↑ Trudeau, Richard J. (1993).
*Introduction to Graph Theory*(Corrected, enlarged republication. ed.). New York: Dover Pub. p. 64. 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.

- ↑ Barthelemy, M. (2017).
*Morphogenesis of Spatial Networks*. New York: Springer. p. 6. - ↑ Schnyder, W. (1989), "Planar graphs and poset dimension",
*Order*,**5**(4): 323–343, doi:10.1007/BF00353652, MR 1010382, S2CID 122785359 . - ↑ Bhasker, Jayaram; Sahni, Sartaj (1988), "A linear algorithm to find a rectangular dual of a planar triangulated graph",
*Algorithmica*,**3**(1–4): 247–278, doi:10.1007/BF01762117, S2CID 2709057 . - ↑ 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 . - ↑ Felsner, Stefan (2004), "1.4 Outerplanar Graphs and Convex Geometric Graphs",
*Geometric graphs and arrangements*, Advanced Lectures in Mathematics, Friedr. Vieweg & Sohn, Wiesbaden, pp. 6–7, doi:10.1007/978-3-322-80303-0_1, ISBN 3-528-06972-4, MR 2061507 - ↑ Sysło, Maciej M.; Proskurowski, Andrzej (1983), "On Halin graphs",
*Graph Theory: Proceedings of a Conference held in Lagów, Poland, February 10–13, 1981*, Lecture Notes in Mathematics,**1018**, Springer-Verlag, pp. 248–256, doi:10.1007/BFb0071635 . - ↑ Giménez, Omer; Noy, Marc (2009). "Asymptotic enumeration and limit laws of planar graphs".
*Journal of the American Mathematical Society*.**22**(2): 309–329. arXiv: math/0501269 . Bibcode:2009JAMS...22..309G. doi:10.1090/s0894-0347-08-00624-3. S2CID 3353537. - ↑ McDiarmid, Colin; Steger, Angelika; Welsh, Dominic J.A. (2005). "Random planar graphs".
*Journal of Combinatorial Theory, Series B*.**93**(2): 187–205. CiteSeerX 10.1.1.572.857 . doi:10.1016/j.jctb.2004.09.007. - ↑ 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. CiteSeerX 10.1.1.106.7456 . doi:10.1007/s00373-006-0647-2. S2CID 22639942. - ↑ 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 - ↑ Bose, Prosenjit; Dujmović, Vida; Javarsineh, Mehrnoosh; Morin, Pat (2020),
*Asymptotically optimal vertex ranking of planar graphs*, arXiv: 2007.06455 - ↑ Dębski, Michał; Felsner, Stefan; Micek, Piotr; Schröder, Felix (2019),
*Improved bounds for centered colorings*, arXiv: 1907.04586 - ↑ I. S. 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.
- ↑ Buhl, J.; Gautrais, J.; Sole, R.V.; Kuntz, P.; Valverde, S.; Deneubourg, J.L.; Theraulaz, G. (2004), "Efficiency and robustness in ant networks of galleries",
*European Physical Journal B*, Springer-Verlag,**42**(1): 123–129, Bibcode:2004EPJB...42..123B, doi:10.1140/epjb/e2004-00364-9, S2CID 14975826 . - ↑ M. Halldórsson, S. Kitaev and A. Pyatkin. Semi-transitive orientations and word-representable graphs, Discr. Appl. Math. 201 (2016), 164-171.
- ↑ T. Z. Q. Chen, S. Kitaev, and B. Y. Sun. Word-representability of face subdivisions of triangular grid graphs, Graphs and Combin. 32(5) (2016), 1749-1761.
- ↑ T. Z. Q. Chen, S. Kitaev, and B. Y. Sun. Word-representability of triangulations of grid-covered cylinder graphs, Discr. Appl. Math. 213 (2016), 60-70.

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 subdivision of *K*_{5} or of *K*_{3,3}.

In graph theory, a branch of mathematics, the (binary) **cycle space** of an undirected graph is the set of its even-degree subgraphs.

This is a **glossary of graph theory**. Graph theory is the study of graphs, systems of nodes or vertices connected in pairs by lines or edges.

In graph theory, an **outerplanar graph** is a graph that has a planar drawing for which all vertices belong to the outer face of the drawing.

In graph theory, two graphs and are **homeomorphic** if there is a graph isomorphism from some **subdivision** of to some **subdivision** of . If the edges of a graph are thought of as lines drawn from one vertex to another, then two graphs are homeomorphic to each other in the graph-theoretic sense precisely if they are homeomorphic in the sense in which the term is used in topology.

In graph theory, an undirected graph *H* is called a **minor** of the graph *G* if *H* can be formed from *G* by deleting edges and vertices and by contracting edges.

In the mathematical discipline of graph theory, the **dual graph** of a plane graph G is a graph that has a vertex for each face of G. The dual graph has an edge for each pair of faces in G that are separated from each other by an edge, and a self-loop when the same face appears on both sides of an edge. Thus, each edge e of G has a corresponding dual edge, whose endpoints are the dual vertices corresponding to the faces on either side of e. The definition of the dual depends on the choice of embedding of the graph G, so it is a property of plane graphs rather than planar graphs. For planar graphs generally, there may be multiple dual graphs, depending on the choice of planar embedding of the graph.

In mathematics, a **dense graph** is a graph in which the number of edges is close to the maximal number of edges. The opposite, a graph with only a few edges, is a **sparse graph**. The distinction between sparse and dense graphs is rather vague, and depends on the context.

In mathematics, **Fáry's theorem** states that any simple planar graph can be drawn without crossings so that its edges are straight line segments. That is, the ability to draw graph edges as curves instead of as straight line segments does not allow a larger class of graphs to be drawn. The theorem is named after István Fáry, although it was proved independently by Klaus Wagner (1936), Fáry (1948), and Sherman K. Stein (1951).

In graph theory, a **cactus** is a connected graph in which any two simple cycles have at most one vertex in common. Equivalently, it is a connected graph in which every edge belongs to at most one simple cycle, or in which every block is an edge or a cycle.

In graph theory, a **peripheral cycle** in an undirected graph is, intuitively, a cycle that does not separate any part of the graph from any other part. Peripheral cycles were first studied by Tutte (1963), and play important roles in the characterization of planar graphs and in generating the cycle spaces of nonplanar graphs.

In graph theory, the **planarity testing** problem is the algorithmic problem of testing whether a given graph is a planar graph. This is a well-studied problem in computer science for which many practical algorithms have emerged, many taking advantage of novel data structures. Most of these methods operate in O(*n*) time, where *n* is the number of edges in the graph, which is asymptotically optimal. Rather than just being a single Boolean value, the output of a planarity testing algorithm may be a planar graph embedding, if the graph is planar, or an obstacle to planarity such as a Kuratowski subgraph if it is not.

The **circle packing theorem** describes the possible tangency relations between circles in the plane whose interiors are disjoint. A **circle packing** is a connected collection of circles whose interiors are disjoint. The intersection graph of a circle packing is the graph having a vertex for each circle, and an edge for every pair of circles that are tangent. If the circle packing is on the plane, or, equivalently, on the sphere, then its intersection graph is called a **coin graph**; more generally, intersection graphs of interior-disjoint geometric objects are called **tangency graphs** or **contact graphs**. Coin graphs are always connected, simple, and planar. The circle packing theorem states that these are the only requirements for a graph to be a coin graph:

In graph theory, the **planar separator theorem** is a form of isoperimetric inequality for planar graphs, that states that any planar graph can be split into smaller pieces by removing a small number of vertices. Specifically, the removal of vertices from an -vertex graph can partition the graph into disjoint subgraphs each of which has at most vertices.

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 its edges cross, and the cycle connects the leaves in their clockwise ordering in this embedding. Thus, the cycle forms the outer face of the Halin graph, with the tree inside it.

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 (simple) 3-vertex-connected planar graphs. That is, every convex polyhedron forms a 3-connected planar graph, and every 3-connected planar graph can be represented as the graph of a convex polyhedron. For this reason, the 3-connected planar graphs are also known as polyhedral graphs. Branko Grünbaum has called this theorem "the most important and deepest known result on 3-polytopes."

In graph theory, a mathematical discipline, coloring refers to an assignment of colours or labels to vertices, edges and faces of a graph. **Defective coloring** is a variant of proper vertex coloring. In a proper vertex coloring, the vertices are coloured such that no adjacent vertices have the same colour. In defective coloring, on the other hand, vertices are allowed to have neighbours of the same colour to a certain extent.

In graph drawing and geometric graph theory, a **Tutte embedding** or **barycentric embedding** of a simple 3-vertex-connected planar graph is a crossing-free straight-line embedding with the properties that the outer face is a convex polygon and that each interior vertex is at the average of its neighbors' positions. If the outer polygon is fixed, this condition on the interior vertices determines their position uniquely as the solution to a system of linear equations. Solving the equations geometrically produces a **planar embedding**. **Tutte's spring theorem**, proven by W. T. Tutte (1963), states that this unique solution is always crossing-free, and more strongly that every face of the resulting planar embedding is convex. It is called the spring theorem because such an embedding can be found as the equilibrium position for a system of springs representing the edges of the graph.

In graph theory, a ** k-outerplanar graph** is a planar graph that has a planar embedding in which the vertices belong to at most concentric layers. The

In graph drawing, a **convex drawing** of a planar graph is a drawing that represents the vertices of the graph as points in the Euclidean plane and the edges as straight line segments, in such a way that all of the faces of the drawing have a convex boundary. The boundary of a face may pass straight through one of the vertices of the graph without turning; a **strictly convex drawing** asks in addition that the face boundary turns at each vertex. That is, in a strictly convex drawing, each vertex of the graph is also a vertex of each convex polygon describing the shape of each incident face.

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*Fundamenta Mathematicae*(in French),**15**: 271–283, doi: 10.4064/fm-15-1-271-283 . - Wagner, K. (1937), "Über eine Eigenschaft der ebenen Komplexe",
*Mathematische Annalen*(in German),**114**: 570–590, doi:10.1007/BF01594196, S2CID 123534907 . - Boyer, John M.; Myrvold, Wendy J. (2005), "On the cutting edge: Simplified O(n) planarity by edge addition" (PDF),
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*International Journal of Foundations of Computer Science*,**17**(5): 1017–1029, arXiv: math/0610935 , doi:10.1142/S0129054106004248, S2CID 40107560 . Special Issue on Graph Drawing. - D.A. Bader and S. Sreshta, A New Parallel Algorithm for Planarity Testing, UNM-ECE Technical Report 03-002, October 1, 2003.
- Fisk, Steve (1978), "A short proof of Chvátal's watchman theorem",
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Wikimedia Commons has media related to 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 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

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