Category: Planar graphs

Medial graph
In the mathematical discipline of graph theory, the medial graph of plane graph G is another graph M(G) that represents the adjacencies between edges in the faces of G. Medial graphs were introduced i
Grinberg's theorem
In graph theory, Grinberg's theorem is a necessary condition for a planar graph to contain a Hamiltonian cycle, based on the lengths of its face cycles. If a graph does not meet this condition, it is
Tutte embedding
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 tha
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
Planarization
In the 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 the non-planar graphs within a la
Friendship graph
In 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 3n edges. The friendship graph Fn can be co
Graphic matroid
In the mathematical theory of matroids, a graphic matroid (also called a cycle matroid or polygon matroid) is a matroid whose independent sets are the forests in a given finite undirected graph. The d
Left-right planarity test
In graph theory, a branch of mathematics, the left-right planarity testor de Fraysseix–Rosenstiehl planarity criterion is a characterization of planar graphs based on the properties of the depth-first
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. Schein
Outerplanar graph
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 graphs may be characterized (analogously to Wagne
26-fullerene graph
In the mathematical field of graph theory, the 26-fullerene graph is a polyhedral graph with V = 26 vertices and E = 39 edges. Its planar embedding has three hexagonal faces (including the one shown a
Dodecahedron
In geometry, a dodecahedron (Greek δωδεκάεδρον, from δώδεκα dōdeka "twelve" + ἕδρα hédra "base", "seat" or "face") or duodecahedron is any polyhedron with twelve flat faces. The most familiar dodecahe
Truncated icosidodecahedron
In geometry, the truncated icosidodecahedron is an Archimedean solid, one of thirteen convex, isogonal, non-prismatic solids constructed by two or more types of regular polygon faces. It has 62 faces:
Subhamiltonian graph
In graph theory and graph drawing, a subhamiltonian graph is a subgraph of a planar Hamiltonian graph.
St-planar 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 face of the graph. That is, it is a directed gr
In the 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 the Cartesian product of two path gr
Planarity testing
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 be drawn in the plane without edge intersections).
Regular icosahedron
In geometry, a regular icosahedron (/ˌaɪkɒsəˈhiːdrən, -kə-, -koʊ-/ or /aɪˌkɒsəˈhiːdrən/) is a convex polyhedron with 20 faces, 30 edges and 12 vertices. It is one of the five Platonic solids, and the
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
Errera graph
In the mathematical field of graph theory, the Errera graph is a graph with 17 vertices and 45 edges. Alfred Errera published it in 1921 as a counterexample to Kempe's erroneous proof of the four colo
Lattice graph
In graph theory, a lattice graph, mesh graph, or grid graph is a graph whose drawing, embedded in some Euclidean space , forms a regular tiling. This implies that the group of bijective transformation
Harborth's conjecture
In mathematics, Harborth's conjecture states that every planar graph has a planar drawing in which every edge is a straight segment of integer length. This conjecture is named after Heiko Harborth, an
Contact graph
In the mathematical area of graph theory, a contact graph or tangency graph is a graph whose vertices are represented by geometric objects (e.g. curves, line segments, or polygons), and whose edges co
Tutte graph
In the mathematical field of graph theory, the Tutte graph is a 3-regular graph with 46 vertices and 69 edges named after W. T. Tutte. It has chromatic number 3, chromatic index 3, girth 4 and diamete
Antiprism graph
In the mathematical field of graph theory, an antiprism graph is a graph that has one of the antiprisms as its skeleton. An n-sided antiprism has 2n vertices and 4n edges. They are regular, polyhedral
Baker's technique
In theoretical computer science, Baker's technique is a method for designing polynomial-time approximation schemes (PTASs) for problems on planar graphs. It is named after Brenda Baker, who announced
Penny graph
In geometric graph theory, a penny graph is a contact graph of unit circles. It is formed from a collection of unit circles that do not cross each other, by creating a vertex for each circle and an ed
Herschel graph
In graph theory, a branch of mathematics, the Herschel graph is a bipartite undirected graph with 11 vertices and 18 edges. It is a polyhedral graph (the graph of a convex polyhedron), and is the smal
Frucht graph
In the mathematical field of graph theory, the Frucht graph is a cubic graph with 12 vertices, 18 edges, and no nontrivial symmetries. It was first described by Robert Frucht in 1939. The Frucht graph
Golomb graph
In graph theory, the Golomb graph is a polyhedral graph with 10 vertices and 18 edges. It is named after Solomon W. Golomb, who constructed it (with a non-planar embedding) as a unit distance graph th
Nested triangles graph
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 vertices on consecutive triangles in the seque
Dürer graph
In the mathematical field of graph theory, the Dürer graph is an undirected graph with 12 vertices and 18 edges. It is named after Albrecht Dürer, whose 1514 engraving Melencolia I includes a depictio
Barnette's conjecture
Barnette's conjecture is an unsolved problem in graph theory, a branch of mathematics, concerning Hamiltonian cycles in graphs. It is named after , a professor emeritus at the University of California
Hanani–Tutte theorem
In topological graph theory, the Hanani–Tutte theorem is a result on the parity of edge crossings in a graph drawing. It states that every drawing in the plane of a non-planar graph contains a pair of
Planarity
Planarity is a puzzle computer game by John Tantalo, based on a concept by Mary Radcliffe at Western Michigan University.The name comes from the concept of planar graphs in graph theory; these are gra
Platonic graph
In the mathematical field of graph theory, a Platonic graph is a graph that has one of the Platonic solids as its skeleton. There are 5 Platonic graphs, and all of them are regular, polyhedral (and th
Truncated tetrahedron
In geometry, the truncated tetrahedron is an Archimedean solid. It has 4 regular hexagonal faces, 4 equilateral triangle faces, 12 vertices and 18 edges (of two types). It can be constructed by trunca
Book (graph theory)
In graph theory, a book graph (often written ) may be any of several kinds of graph formed by multiple cycles sharing an edge.
Fáry's theorem
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 line segments. That is, the ability to draw
Kittell graph
In the mathematical field of graph theory, the Kittell graph is a planar graph with 23 vertices and 63 edges. Its unique planar embedding has 42 triangular faces. The Kittell graph is named after Irvi
Barnette–Bosák–Lederberg graph
In the mathematical field of graph theory, the Barnette–Bosák–Lederberg graph is a cubic (that is, 3-regular) polyhedral graph with no Hamiltonian cycle, the smallest such graph possible. It was disco
K-outerplanar 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 outerplanarity index of a planar graph is the minimu
Apex graph
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 an apex of the graph. It is an apex, not the
Wiener–Araya graph
The Wiener–Araya graph is, in graph theory, a graph on 42 vertices with 67 edges. It is hypohamiltonian, which means thatit does not itself have a Hamiltonian cycle but every graph formed by removing
Kelmans–Seymour conjecture
In graph theory, the Kelmans–Seymour conjecture states that every 5-vertex-connected graph that is not planar contains a subdivision of the 5-vertex complete graph K5. It is named for Paul Seymour and
Bidiakis cube
In the mathematical field of graph theory, the Bidiakis cube is a 3-regular graph with 12 vertices and 18 edges.
Dual graph
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 separ
Good spanning tree
In the mathematical field of graph theory, a good spanning tree of an embedded planar graph is a rooted spanning tree of whose non-tree edges satisfy the following conditions. * there is no non-tree
Mac Lane's planarity criterion
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 published it in 1937. It states that a finite
FKT algorithm
The FKT algorithm, named after Fisher, Kasteleyn, and Temperley, counts the number of perfect matchings in a planar graph in polynomial time. This same task is #P-complete for general graphs. For matc
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
Circle packing theorem
The circle packing theorem (also known as the Koebe–Andreev–Thurston theorem) describes the possible tangency relations between circles in the plane whose interiors are disjoint. A circle packing is a
Cactus graph
In graph theory, a cactus (sometimes called a cactus tree) 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 e
Tait's conjecture
In mathematics, Tait's conjecture states that "Every 3-connected planar cubic graph has a Hamiltonian cycle (along the edges) through all its vertices". It was proposed by P. G. Tait and disproved by
Convex drawing
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
Goldner–Harary graph
In the mathematical field of graph theory, the Goldner–Harary graph is a simple undirected graph with 11 vertices and 27 edges. It is named after A. Goldner and Frank Harary, who proved in 1975 that i
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
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 b
Schnyder's theorem
In graph theory, Schnyder's theorem is a characterization of planar graphs in termsof the order dimension of their incidence posets. It is named after Walter Schnyder, who published its proof in . The
Eodermdrome
An eodermdrome is a form of word play wherein a word (or phrase) is formed from a set of letters (or words) in such a way that it has a non-planar spelling net. Gary S. Bloom, Allan Gewirtz, John W. K
Planar separator theorem
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
Moser spindle
In graph theory, a branch of mathematics, the Moser spindle (also called the Mosers' spindle or Moser graph) is an undirected graph, named after mathematicians Leo Moser and his brother William, with
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 gra
Thickness (graph theory)
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 a collection of k planar graphs, all having th
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
Planar graph
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
Polyhedral graph
In geometric graph theory, a branch of mathematics, a polyhedral graph is the undirected graph formed from the vertices and edges of a convex polyhedron. Alternatively, in purely graph-theoretic terms
Grötzsch's theorem
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. According to the four-color theorem, every g
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 pla
Map graph
In graph theory, a branch of mathematics, a map graph is an undirected graph formed as the intersection graph of finitely many simply connected and internally disjoint regions of the Euclidean plane.
Series–parallel graph
In graph theory, series–parallel graphs are graphs with two distinguished vertices called terminals, formed recursively by two simple composition operations. They can be used to model series and paral
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
Walther graph
In the mathematical field of graph theory, the Walther graph, also called the Tutte fragment, is a planar bipartite graph with 25 vertices and 31 edges named after Hansjoachim Walther. It has chromati
Hanoi graph
In graph theory and recreational mathematics, the Hanoi graphs are undirected graphs whose vertices represent the possible states of the Tower of Hanoi puzzle, and whose edges represent allowable move
Upward planar drawing
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 as non-crossing monotonic upwards curve
Bull graph
In the mathematical field of graph theory, the bull graph is a planar undirected graph with 5 vertices and 5 edges, in the form of a triangle with two disjoint pendant edges. It has chromatic number 3
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
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 d
Regular dodecahedron
A regular dodecahedron or pentagonal dodecahedron is a dodecahedron that is regular, which is composed of 12 regular pentagonal faces, three meeting at each vertex. It is one of the five Platonic soli
Kotzig's theorem
In graph theory and polyhedral combinatorics, areas of mathematics, Kotzig's theorem is the statement that every polyhedral graph has an edge whose two endpoints have total degree at most 13. An extre
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 c
Whitney's planarity criterion
In mathematics, 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 only if its graphic matro
Prism graph
In the mathematical field of graph theory, a prism graph is a graph that has one of the prisms as its skeleton.
Squaregraph
In graph theory, a branch of mathematics, a squaregraph is a type of undirected graph that can be drawn in the plane in such a way that every bounded face is a quadrilateral and every vertex with thre
Strangulated graph
In graph theoretic mathematics, a strangulated graph is a graph in which deleting the edges of any induced cycle of length greater than three would disconnect the remaining graph. That is, they are th
1-planar graph
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 point, where it crosses a single additional edge
Wheel graph
In the mathematical discipline of graph theory, a wheel graph is a graph formed by connecting a single universal vertex to all vertices of a cycle. A wheel graph with n vertices can also be defined as
Blossom tree (graph theory)
In the study 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 trees can be used to sample ran
Archimedean graph
In the mathematical field of graph theory, an Archimedean graph is a graph that forms the skeleton of one of the Archimedean solids. There are 13 Archimedean graphs, and all of them are regular, polyh