Definition: A graph is considered Planar if it can be redrawn such that no edges intersect. That is, a graph is planar if there exists a plane drawing of the graph. In many instances, we may want to see if a graph can be redrawn without any edge intersection using what is called a Plane Drawing.  For example, the following graph is considered planar: Because there exists a plane drawing of the graph, that is, a reconfiguration of the location of the edges so that no edges in the graph intersect. Some graphs are not planar though since there is no possible plane drawing of the graph, for example, the complete bipartite graph $K_{3,3}$: is not considered planar since there must be at least one edge intersection in the graph. Planar Graph Example, Properties & Practice Problems are discussed.  A graph is a collection of vertices connected to each other through a set of edges. The study of graphs is known as Graph Theory. In this article, we will discuss about Planar Graphs. Planar Graph-. A planar graph may be defined as-. In graph theory, Planar graph is a graph that can be drawn in a plane such that none of its edges cross each other. Planar Graph Example-. The following graph is an example of a planar graph-. Here, In this graph, no two edges cross each other. Therefore, What Are Planar Graphs Used For it is a planar graph. Regions of Plane-. The planar representation of the graph splits the plane into connecte. Planar — can refer to:* Planar (computer graphics), computer graphics pixel information from several bitplanes * Planar graph, graph that can be drawn so that no edges intersect (or that can be embedded) in the plane * Plane (mathematics), a name for Wikipedia. Graph coloring — A proper vertex coloring of the Petersen graph with 3 colors, the minimum number possible. In graph theory, graph coloring is a special case of graph labeling; it is an assignment of labels traditionally called colors to elements of a graph Wikipedia. graph theory — Math. the branch of mathematics dealing with lin. Therefore, by Theorem 2, it cannot be planar. Alternating Planar Graphs. There are a number of measures characterizing the degree by which a graph fails to be planar, among these being the graph crossing numberrectilinear crossing numberplanar graph是什么 skewnessgraph coarsenessand graph thickness. Semi-transitive orientations grwph是什么 word-representable graphs, Discr. Plsnar corresponding numbers of planar planar graph是什么 graphs are 1, 1, 1, 2, 6, 20, 99,Reading, MA: Addison-Wesley, pp.

Auslander, L. Mechanics 10 , , Battle, J. Booth, K. System Sci. Bryant, V. Cai, J. New York University, Di Battista, G. Graph Drawing: Algorithms for the Visualization of Graphs. Eades, P. Department of Computer Science. Eppstein, D. Even, S. Graph Algorithms.

Szeged 11 , , Gardner, M. Harary, F. Reading, MA: Addison-Wesley, pp. Harborth, H. Hopcroft, J. ACM 21 , , Kocay, W. Le Lionnais, F. Les nombres remarquables. Paris: Hermann, p. Nishizeki, T. Planar Graph Drawing. Singapore: World Scientific, Read, R. Scheinerman, E. Monthly , , Skiena, S. Sloane, N. Steinbach, P.

Field Guide to Simple Graphs. Albuquerque, NM: Design Lab, Stony Brook Algorithm Repository. Wagon, S. New York: Springer-Verlag, pp. West, D. Introduction to Graph Theory, 2nd ed. Whitney, H. Wilson, R. Introduction to Graph Theory. London: Longman, 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. 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.

Euler's formula can also be proved as follows: if the graph isn't a tree , then remove an edge which completes a cycle.

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.

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 , states that a graph is planar if and only if it is a coin graph. 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.

A simple graph is called Combination Lock For Kitchen Cabinet Graph 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. 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.

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.

A 1-outerplanar embedding of a graph is the same as an 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.

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 Shelf Mounting Hardware Hidden Graph 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 Soft Close Drawer Slides How Does It Work Graph 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.

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. Almost all planar graphs have an exponential number of automorphisms. The Four Color Theorem states that every planar graph is 4- colorable i.

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. 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. 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.

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. Thus, it ranges from 0 for trees to 1 for maximal planar graphs.

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