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Graph that can be embedded in the plane
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
Planar_graph
Graph with at most one crossing per edge
In topological graph theory, a 1-planar graph is a graph that can be drawn in the Euclidean plane in such a way that each edge has at most one crossing
1-planar_graph
Graph representing faces of another graph
mathematical discipline of graph theory, the dual graph of a planar graph G is a graph that has a vertex for each face of G. The dual graph has an edge for each
Dual_graph
Non-crossing graph with vertices on outer face
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. Outerplanar
Outerplanar_graph
Planar maps require at most four colors
terms of graph theory, by considering it in terms of constructing a graph coloring of the planar graph of adjacencies between regions. In graph-theoretic
Four_color_theorem
Planar directed acyclic graph
In graph theory, an st-planar graph is a bipolar orientation of a plane graph for which both the source and the sink of the orientation are on the outer
St-planar_graph
Cycle graph plus universal vertex
every wheel graph is a Halin graph. They are self-dual: the planar dual of any wheel graph is an isomorphic graph. Every maximal planar graph, other than
Wheel_graph
Area of discrete mathematics
straight-line graph. Any planar graph can be represented as a planar straight-line graph by Fáry's theorem. The planar straight-line graph is the special
Graph_theory
Algorithmic problem of finding non-crossing drawings
In graph theory, the planarity testing problem is the algorithmic problem of testing whether a given graph is a planar graph (that is, whether it can
Planarity_testing
3-regular graph with no 3-edge-coloring
equivalent forms of the four color theorem is that every snark is a non-planar graph. Research on snarks originated in Peter G. Tait's work on the four color
Snark_(graph_theory)
Subgraph with contracted edges
The theory of graph minors began with Wagner's theorem that a graph is planar if and only if its minors include neither the complete graph K5 nor the complete
Graph_minor
Graph layout on multiple half-planes
In graph theory, a book embedding is a generalization of planar embedding of a graph to embeddings in a book, a collection of half-planes all having the
Book_embedding
Undirected unit-distance graph requiring four colors
matchstick graph. The Moser spindle is also a Laman graph, meaning that it forms a minimally rigid system when embedded in the plane. As a planar Laman graph, it
Moser_spindle
Planar graph embedding where edges map to straight-line segments
graph theory, a planar straight-line graph (PSLG), also called a straight-line plane graph or plane straight-line graph, is an embedding of a planar graph
Planar_straight-line_graph
Topics referred to by the same term
semiconductor devices, such as planar transistors Planar graph, graph that can be drawn in the plane so that no edges cross Planar mechanism, a system of parts
Planar
Number of planar subgraphs to cover a graph
In graph theory, the thickness of a graph G is the minimum number of planar graphs into which the edges of G can be partitioned. That is, if there exists
Thickness_(graph_theory)
Describing a family of graphs by excluding certain (sub)graphs
graph is planar (can be drawn without crossings in the plane) if and only if it does not contain either of two forbidden graphs, the complete graph K5
Forbidden graph characterization
Forbidden_graph_characterization
Graph made from vertices and edges of a convex polyhedron
polyhedron. Alternatively, in purely graph-theoretic terms, the polyhedral graphs are the 3-vertex-connected, planar graphs. The analogue concept for polytopes
Polyhedral_graph
Methodic assignment of colors to elements of a graph
no two adjacent edges are of the same color, and a face coloring of a planar graph assigns a color to each face (or region) so that no two faces that share
Graph_coloring
Any planar graph can be subdivided by removing a few vertices
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
Planar_separator_theorem
On coloring the edges of graphs
path of two adjacent edges. In Vizing's planar graph conjecture, Vizing (1965) states that all simple, planar graphs with maximum degree six or seven are
Vizing's_theorem
Graph whose embedding in a Euclidean space forms a regular tiling
In graph theory, a lattice graph, mesh graph, or grid graph is a graph whose drawing, embedded in some Euclidean space R n {\displaystyle \mathbb {R}
Lattice_graph
Decomposition of a graph into hamiltonion cycles
3-regular graph is planar and bipartite, when it is a Halin graph, when it is itself a prism or Möbius ladder, or when it is a generalized Petersen graph of
Hamiltonian_decomposition
Vertices connected in pairs by edges
vertices is 1. If a path graph occurs as a subgraph of another graph, it is a path in that graph. A planar graph is a graph whose vertices and edges can
Graph_(discrete_mathematics)
Graph of king moves on a chessboard
graph of small chessboards, other drawings lead to even fewer crossings; in particular every 2 × n {\displaystyle 2\times n} king's graph is a planar
King's_graph
Graph of short distances in another graph
and to find graph drawings with high angular resolution. Both the chromatic number and the degeneracy of the kth power of a planar graph of maximum degree
Graph_power
Graph of triangles with a shared vertex
the mathematical field of graph theory, the friendship graph (or Dutch windmill graph or n-fan) Fn is a planar, undirected graph with 2n + 1 vertices and
Friendship_graph
Mathematical puzzle of avoiding crossings
a graph embedding in the plane. The impossibility of the puzzle corresponds to the fact that K 3 , 3 {\displaystyle K_{3,3}} is not a planar graph. Multiple
Three_utilities_problem
apex graph is a graph in which one vertex can be removed, leaving a planar subgraph. The removed vertex is called the apex. A k-apex graph is a graph that
Glossary_of_graph_theory
On cycle bases of planar graphs
In graph theory, Mac Lane's planarity criterion is a characterisation of planar graphs in terms of their cycle spaces, named after Saunders Mac Lane who
Mac Lane's planarity criterion
Mac_Lane's_planarity_criterion
it was named after Errera by Hutchinson & Wagon (1998). The Errera graph is planar and has chromatic number 4, chromatic index 6, radius 3, diameter 4
Errera_graph
All even-degree subgraphs of a graph
undirected graph is planar if and only if the graph has a cycle basis in which each edge of the graph participates in at most two basis cycles. In a planar graph
Cycle_space
Unproven conjecture in graph theory
searches found four graphs on 24 vertices in which the only power-of-two cycles have 16 vertices. One of these four graphs is planar; however, the Erdős–Gyárfás
Erdős–Gyárfás_conjecture
Graph related to another graph by a covering map
(hypothetical) crystals. A planar cover of a graph is a finite covering graph that is itself a planar graph. The property of having a planar cover may be characterized
Covering_graph
4-regular undirected graph in mathematics
thickness 3 and queue number 2. The graph is neither planar nor 1-planar. The Robertson graph is also a Hamiltonian graph which possesses 5,376 distinct directed
Robertson_graph
Embedding a graph in 3D space with no cycles interlinked
graph. A linklessly embeddable graph is a graph that has a linkless or flat embedding; these graphs form a three-dimensional analogue of the planar graphs
Linkless_embedding
On graph drawing with integer edge lengths
every planar graph have an integral Fáry embedding? More unsolved problems in mathematics In mathematics, Harborth's conjecture states that every planar graph
Harborth's_conjecture
Graph theory concept
In graph theory, a planar cover of a finite graph G is a finite covering graph of G that is itself a planar graph. Every graph that can be embedded into
Planar_cover
Embedding of the circle in three dimensional Euclidean space
at these crossings. In graph theory terms, a regular projection of a knot, or knot diagram is thus a quadrivalent planar graph with over/under-decorated
Knot_(mathematics)
complete graph K4 (such a characterisation is known for K4-free planar graphs) Classify graphs with representation number 3, that is, graphs that can
List of unsolved problems in mathematics
List_of_unsolved_problems_in_mathematics
Triangle-free graph requiring four colors
theorem that planar triangle-free graphs are 3-colorable. The Grötzsch graph is a member of an infinite sequence of triangle-free graphs, each the Mycielskian
Grötzsch_graph
On tangency patterns of circles
be a coin graph: every finite connected simple planar graph G {\displaystyle G} has a circle packing in the plane whose intersection graph is isomorphic
Circle_packing_theorem
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
Meshedness_coefficient
Path in a graph that visits each vertex exactly once
vertices in the whole graph. Theorem—A 4-connected planar triangulation has a Hamiltonian cycle. Theorem—A 4-connected planar graph has a Hamiltonian cycle
Hamiltonian_path
Planar graph with 5 nodes and 6 edges
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
Butterfly_graph
Unsolved problem in computational complexity theory
Bounded-parameter graphs Graphs of bounded treewidth Graphs of bounded genus (Planar graphs are graphs of genus 0.) Graphs of bounded degree Graphs with bounded
Graph_isomorphism_problem
Graph where every edge is in one triangle
to planar graphs. Let G {\displaystyle G} be a planar graph embedded in the plane in such a way that every face is a quadrilateral, such as the graph of
Locally_linear_graph
Undirected graph with 11 nodes and 27 edges
proved in 1975 that it was the smallest non-Hamiltonian maximal planar graph. The same graph had already been given as an example of a non-Hamiltonian simplicial
Goldner–Harary_graph
On forbidden subgraphs in planar graphs
In graph theory, Kuratowski's theorem is a mathematical forbidden graph characterization of planar graphs, named after Kazimierz Kuratowski. It states
Kuratowski's_theorem
Intersection graph representing regions on the Euclidean plane
internally disjoint regions of the Euclidean plane. The map graphs include the planar graphs, but are more general. Any number of regions can meet at a
Map_graph
Geometric graph with unit edge lengths
distance graphs. Matchstick graphs are a special case of unit distance graphs, in which no edges cross. Every matchstick graph is a planar graph, but some
Unit_distance_graph
Study of graphs defined by geometric means
theorem states that any planar graph may be represented as a planar straight line graph. A triangulation is a planar straight line graph to which no more edges
Geometric_graph_theory
Planar graph used as counterexample
In graph theory, a nested triangles graph with n vertices is a planar graph formed from a sequence of n/3 triangles, by connecting pairs of corresponding
Nested_triangles_graph
Characterization of planar graphs by matroids
Whitney's planarity criterion is a matroid-theoretic characterization of planar graphs, named after Hassler Whitney. It states that a graph G is planar if and
Whitney's_planarity_criterion
Graph with almost the max amount of edges
is planar, together imply that the planar graphs are (3,6)-sparse. However, not every (3,6)-sparse graph is planar. Similarly, outerplanar graphs are
Dense_graph
Visualization of node-link graphs
the graph is planar, then it is often convenient to draw it without any edge intersections; that is, in this case, a graph drawing represents a graph embedding
Graph_drawing
Points usable to draw any planar graph
problem in mathematics Do planar graphs have universal point sets of subquadratic size? More unsolved problems in mathematics In graph drawing, a universal
Universal_point_set
Graph-theoretic description of polyhedra
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
Steinitz's_theorem
Graph with an antiprism as its skeleton
vertex-transitive, and planar graphs), and also Hamiltonian graphs. The first graph in the sequence, the tetrahedral graph, has 4 verticies and 6 edges
Antiprism_graph
Graph which can be made planar by removing a single node
In graph theory, a branch of mathematics, an apex graph is a graph that can be made planar by the removal of a single vertex. The deleted vertex is called
Apex_graph
Technique for drawing non-planar graphs
mathematical field of graph theory, planarization is a method of extending graph drawing methods from planar graphs to graphs that are not planar, by embedding
Planarization
Mathematical tree of cycles
in any graph may be found in polynomial time using an algorithm for the matroid parity problem. Since triangular cactus graphs are planar graphs, the largest
Cactus_graph
Algorithm for counting perfect matchings in planar graphs
number of perfect matchings in a planar graph in polynomial time. This same task is #P-complete for general graphs. For matchings that are not required
FKT_algorithm
Intersection graph for curves in the plane
string graphs was eventually proven to be NP-complete, implying that no simple characterization is likely to exist. Every planar graph is a string graph: one
String_graph
Graph drawing with vertices on a line
maximal planar graph such as the Goldner–Harary graph cannot have a planar embedding with one semicircle per edge. Testing whether a given graph has a crossing-free
Arc_diagram
Puzzle computer game involving planar graphs
planar graphs in graph theory; these are graphs that can be embedded in the Euclidean plane so that no edges intersect. By Fáry's theorem, if a graph
Planarity
planar structures. However, this characterization, the Geiringer–Laman theorem, had already been discovered in 1927 by Hilda Geiringer. Laman graphs arise
Laman_graph
Type of planar 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 k {\displaystyle k} concentric
K-outerplanar_graph
Fewest edge crossings in drawing of a graph
a graph is planar if and only if its crossing number is zero. Determining the crossing number continues to be of great importance in graph drawing, as
Crossing number (graph theory)
Crossing_number_(graph_theory)
Graph with an Archimedean solid as its skeleton
3-vertex-connected planar graphs), and also Hamiltonian graphs. Along with the 13, the infinite sets of prism graphs and antiprism graphs can also be considered
Archimedean_graph
Bipartite non-Hamiltonian polyhedral graph
three of the six red vertices. The Herschel graph is a polyhedral graph; this means that it is a planar graph, one that can be drawn in the plane with none
Herschel_graph
decomposing planar graphs. A k-separator (or k-cutset) in a k-connected graph is a subset of k vertices whose deletion disconnects the remaining graph. For planar
Laminar_set_family
Planar graph with convex polygon faces
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
Convex_drawing
Theorem relating graph minors and topological embeddings
applies if H is a planar graph, and both reasons apply if H is not planar. We first make precise these notions. The tree width of a graph G is a positive
Graph_structure_theorem
Planar graph with 4 nodes and 5 edges
mathematical field of graph theory, the diamond graph is a planar, undirected graph with 4 vertices and 5 edges. It consists of a complete graph K 4 {\displaystyle
Diamond_graph
Subgraph of planar graph with Hamiltonian cycle
In graph theory and graph drawing, a subhamiltonian graph is a subgraph of a planar Hamiltonian graph. A graph G is subhamiltonian if G is a subgraph
Subhamiltonian_graph
Graph with edges non-crossing and upward
In graph drawing, an upward planar drawing of a directed acyclic graph is an embedding of the graph into the Euclidean plane, in which the edges are represented
Upward_planar_drawing
Embedding a graph in a topological space, often Euclidean
known that any finite graph can be embedded in 3-dimensional Euclidean space R 3 {\displaystyle \mathbb {R} ^{3}} . A planar graph is one that can be embedded
Graph_embedding
Concept in graph theory
In graph theory, a nowhere-zero flow or NZ flow is a network flow that is nowhere zero. It is intimately connected (by duality) to coloring planar graphs
Nowhere-zero_flow
Trees with additional directed half edges
of planar graphs, blossom trees are trees with additional directed half edges. Each blossom tree is associated with an embedding of a planar graph. Blossom
Blossom_tree_(graph_theory)
Graph containing cycles of all possible lengths
wheel graphs. A maximal planar graph is a planar graph in which all faces, even the outer face, are triangles. A maximal planar graph is node-pancyclic if
Pancyclic_graph
Graph representing intersections between given sets
states that every planar graph can also be represented as an intersection graph of line segments in the plane. However, intersection graphs of line segments
Intersection_graph
Superconformal Yang–Mills theory
0 (planar graph) contribution survives. Planar Feynman diagrams are graphs in which no propagator cross over another one, in contrast to non-planar Feynman
N = 4 supersymmetric Yang–Mills theory
N_=_4_supersymmetric_Yang–Mills_theory
Graph representing edges of another graph
In the mathematical discipline of graph theory, the line graph of an undirected graph G is another graph L(G) that represents the adjacencies between edges
Line_graph
Every triangle-free planar graph is 3-colorable
In the mathematical field of graph theory, Grötzsch's theorem is the statement that every triangle-free planar graph can be colored with only three colors
Grötzsch's_theorem
Length of a shortest cycle contained in the graph
the graph is planar. In terms of lower bounds, computing the girth of a graph is at least as hard as solving the triangle finding problem on the graph. R
Girth_(graph_theory)
Cycles in a graph that generate all cycles
embedding of the graph forms a cycle basis. The minimum weight cycle basis of a planar graph corresponds to the Gomory–Hu tree of the dual graph. A spanning
Cycle_basis
Graph formed by subdivision of triangles
equivalently be defined as the planar 3-trees, the maximal planar chordal graphs, the uniquely 4-colorable planar graphs, and the graphs of stacked polytopes.
Apollonian_network
Planar, undirected graph with 2n vertices and 3n-2 edges
mathematical field of graph theory, the ladder graph Ln is a planar, undirected graph with 2n vertices and 3n − 2 edges. The ladder graph can be obtained as
Ladder_graph
be no Hamiltonian cycle. The resulting graph is 3-connected and planar, so by Steinitz' theorem it is the graph of a polyhedron. It has 25 faces. It can
Tutte_graph
Least-weight tree connecting graph vertices
problem for planar graphs in linear time. By the Euler characteristic of planar graphs, m ≤ 3n - 6 ∈ O(n), so this is in time O(n). Given graph G where the nodes
Minimum_spanning_tree
Branch of the mathematical field of graph theory
theorem. Crossing number (graph theory) Genus Planar graph Real tree Toroidal graph Topological combinatorics Voltage graph Gross, J.L.; Tucker, T.W.
Topological_graph_theory
Mathematical tree with cycle through leaves
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
Halin_graph
Planar graph drawn by relaxing springs
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
Tutte_embedding
Binary operation in graph theory
king's graph, the graph of moves of a chess king on a chessboard, which can be constructed as a strong product of path graphs. Decompositions of planar graphs
Strong_product_of_graphs
Disproven graph theory
In mathematics, Tait's conjecture states that "Every 3-connected planar cubic graph has a Hamiltonian cycle (along the edges) through all its vertices"
Tait's_conjecture
Planar graphs have straight drawings
In the mathematical field of graph theory, Fáry's theorem states that any simple, planar graph can be drawn without crossings so that its edges are straight
Fáry's_theorem
On Hamiltonian cycles in planar graphs
In graph theory, a theorem of W. T. Tutte states that every 4-vertex-connected planar graph has a Hamiltonian cycle. It strengthens an earlier theorem
Tutte's theorem on Hamiltonian cycles
Tutte's_theorem_on_Hamiltonian_cycles
Computational problem
remains to show that planar GG is PSPACE-hard. This can be proved by showing how to convert an arbitrary graph into a planar graph, such that a game of
Generalized_geography
General concept and operation in mathematics
polyhedron, one can form a planar graph, the graph of its vertices and edges. The dual polyhedron has a dual graph, a graph with one vertex for each face
Duality_(mathematics)
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