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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
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
General concept and operation in mathematics
dual. The dual graph depends on how the primal graph is embedded: different planar embeddings of a single graph may lead to different dual graphs. Matroid
Duality_(mathematics)
Characterization of planar graphs by matroids
graphs, named after Hassler Whitney. It states that a graph G is planar if and only if its graphic matroid is also cographic (that is, it is the dual
Whitney's_planarity_criterion
Procedures for constructing new graphs in graph theory
dual graph; medial graph; quotient graph; double graph; simplex graph; YΔ- and ΔY-transformation; Mycielskian. Binary operations create a new graph from
Graph_operations
Partition of a simple polygon into triangles
useful graph that is often associated with a triangulation of a polygon P is the dual graph. Given a triangulation TP of P, one defines the graph G(TP)
Polygon_triangulation
Shape made from cubes joined together
from the similarly-named notions of a dual polyhedron, and of the dual graph of a surface-embedded graph. Dual graphs have also been used to define and study
Polycube
Polyhedron associated with another by swapping vertices for faces
form a graph embedded on this surface, and the vertices and edges of the (abstract) dual polyhedron form the dual graph of the original graph. An abstract
Dual_polyhedron
On tangency patterns of circles
applies to any polyhedral graph and its dual graph, and proves the existence of a primal–dual packing, circle packings for both graphs that cross at right angles
Circle_packing_theorem
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
Graph representing edges of another graph
used for the line graph include the covering graph, the derivative, the edge-to-vertex dual, the conjugate, the representative graph, and the θ-obrazom
Line_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
Vertices connected in pairs by edges
as: edge contraction, line graph, dual graph, complement graph, graph rewriting; binary operations, which create a new graph from two initial ones, such
Graph_(discrete_mathematics)
Generalization of graph theory
hypergraph is a generalization of a graph in which an edge can join any number of vertices. In contrast, in an ordinary graph, an edge connects exactly two
Hypergraph
Graph whose maximal clique hypergraph is a hypertree
In the mathematical area of graph theory, an undirected graph G is dually chordal if the hypergraph of its maximal cliques is a hypertree. The name comes
Dually_chordal_graph
Form taken by the network of interconnections of a circuit
other. The dual of a graph can be found entirely by a graphical method. The dual of a graph is another graph. For a given tree in a graph, the complementary
Circuit_topology_(electrical)
On bipartite matching and vertex cover
In the mathematical area of graph theory, Kőnig's theorem, proved by Dénes Kőnig (1931), describes an equivalence between the maximum matching problem
Kőnig's theorem (graph theory)
Kőnig's_theorem_(graph_theory)
The join graphs and join trees of a constraint satisfaction problem are graphs representing its dual problem or a problem obtained from the dual problem
Constraint satisfaction dual problem
Constraint_satisfaction_dual_problem
Two special graphs in graph theory
the orientable surface of genus 3, in which they form dual graphs. This is a 3-regular (cubic) graph with 56 vertices and 84 edges, named after Felix Klein
Klein_graphs
Length of a shortest cycle contained in the graph
In graph theory, the girth of an undirected graph is the length of a shortest cycle contained in the graph. If the graph does not contain any cycles (that
Girth_(graph_theory)
Vertex partition in a directed graph
graph, whose total weight is as large as possible. It can be solved in polynomial time. In weakly connected planar graphs, dicuts and cycles are dual
Dicut
Aspect of topological graph theory
In topological graph theory, the Petrie dual of an embedded graph (on a 2-manifold with all faces disks) is another embedded graph that has the Petrie
Petrie_dual
Methodic assignment of colors to elements of a graph
just a vertex coloring of its line graph, and a face coloring of a plane graph is just a vertex coloring of its dual. However, non-vertex coloring problems
Graph_coloring
All even-degree subgraphs of a graph
In graph theory, a branch of mathematics, the (binary) cycle space of an undirected graph is the set of its even-degree spanning subgraphs, or the set
Cycle_space
involved in a constraint is called the constraint scope. The dual constraint graph is the graph in which the vertices are all constraint scopes involved in
Constraint_graph
Graph with nodes connected in a closed chain
In graph theory, a cycle graph or circular graph is a graph that consists of a single cycle, or in other words, some number of vertices (at least 3, if
Cycle_graph
Abstract regular polyhedron with 6 pentagonal faces
of graph theory this is an embedding of the Petersen graph on a real projective plane. With this embedding, the dual graph is K6 (the complete graph with
Hemi-dodecahedron
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
Edges crossing all dicuts in a directed graph
In planar graphs, dijoins and feedback arc sets are dual concepts. The dual graph of a directed graph, embedded in the plane, is a graph with a vertex
Dijoin
Cubic graph with 10 vertices and 15 edges
bridgeless graph has a cycle-continuous mapping to the Petersen graph. More unsolved problems in mathematics In the mathematical field of graph theory, the
Petersen_graph
Appendix:Glossary of graph theory in Wiktionary, the free dictionary. This is a glossary of graph theory. Graph theory is the study of graphs, systems of nodes
Glossary_of_graph_theory
Planar directed acyclic graph
the sink of an st-planar graph back to the source, through the outer face, and then constructs the dual graph (oriented each dual edge clockwise with respect
St-planar_graph
Triangulation method
triangulation of a discrete point set P in general position corresponds to the dual graph of the Voronoi diagram for P. The circumcenters of Delaunay triangles
Delaunay_triangulation
In polytope theory, the edge graph (also known as vertex-edge graph or just graph) of a polytope is a combinatorial graph whose vertices and edges correspond
Graph_of_a_polytope
Graph operation
operation) is a graph operation defined on regular polyhedral graphs with degree 3 or 4. It also applies to the dual graph of these graphs, i.e. graphs with triangular
Goldberg–Coxeter_construction
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
Describing a family of graphs by excluding certain (sub)graphs
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
Forbidden graph characterization
Forbidden_graph_characterization
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
Abstraction of linear independence of vectors
planar graph. In this case, the dual of M {\displaystyle M} is the matroid of the dual graph of G {\displaystyle G} . The dual of a vector matroid representable
Matroid
Mathematical problem
polygon triangulations, we observe that the weak dual graph to the triangulation (the undirected graph having one vertex per triangle and one edge per
Art_gallery_problem
Two closely related models for generating random graphs
the mathematical field of graph theory, the Erdős–Rényi models are two closely related models for generating random graphs and the evolution of a random
Erdős–Rényi_model
Edge-face adjacencies in another graph
medial graph of G and the medial graph of the dual graph of G are isomorphic. Conversely, for any 4-regular plane graph H, the only two plane graphs with
Medial_graph
Chemical graph theory concerns the graph-theoretic structure of molecules and other clusters of atoms. Both the Errera graph itself and its dual graph are
Errera_graph
Mathematical theory on behavior of connected clusters in a random graph
1/2. The dual graph of the square lattice ℤ2 is also the square lattice. It follows that, in two dimensions, the supercritical phase is dual to a subcritical
Percolation_theory
2D graphic with logarithmic scales on both axes
In science and engineering, a log–log graph or log–log plot is a two-dimensional graph of numerical data that uses logarithmic scales on both the horizontal
Log–log_plot
In the mathematical field of graph theory, the Dyck graph is a 3-regular graph with 32 vertices and 48 edges, named after Walther von Dyck. It is Hamiltonian
Dyck_graph
Topics referred to by the same term
Primal graph may refer to: Primal graph (hypergraphs) of a hypergraph A primal graph may be the planar graph from which a dual graph is formed Primal
Primal_graph
Graph generated by a random process
In mathematics, random graph is the general term to refer to probability distributions over graphs. Random graphs may be described simply by a probability
Random_graph
Matroid with graph forests as independent sets
whose independent sets are the forests in a given finite undirected graph. The dual matroids of graphic matroids are called co-graphic matroids or bond
Graphic_matroid
Groot dual Dual abelian variety Dual basis in a field extension Dual bundle Dual curve Dual (category theory) Dual graph Dual group Dual object Dual pair
List_of_dualities
Type of pattern that does not change from one generation to the next
7–11. doi:10.1016/S0167-6377(00)00016-X.. Smith, Barbara M. (2002). "A dual graph translation of a problem in 'Life'". Principles and Practice of Constraint
Still life (cellular automaton)
Still_life_(cellular_automaton)
Association of electrical terms into pairs based on interchanging voltage and current
Y_{c}=Cs} Duality (electricity and magnetism) Duality (mechanical engineering) Dual impedance Dual graph Mechanical–electrical analogies List of dualities Belevitch
Duality_(electrical_circuits)
Non-orientable surface with one edge
of these six regions form Tietze's graph, which is a dual graph on this surface for the six-vertex complete graph but cannot be drawn without crossings
Möbius_strip
Matroid with complemented basis sets
Matroid duals go back to the original paper by Hassler Whitney defining matroids. They generalize to matroids the notions of plane graph duality. Duality is
Dual_matroid
2 player paper and pencil game
played on a triangular grid or a hexagonal grid. Dots and boxes has a dual graph form called "Strings-and-Coins". This game is played on a network of coins
Dots_and_boxes
3-regular graph with no 3-edge-coloring
In the mathematical field of graph theory, a snark is an undirected graph with exactly three edges per vertex whose edges cannot be colored with only three
Snark_(graph_theory)
Symmetric bipartite cubic graph with 16 vertices and 24 edges
In the mathematical field of graph theory, the Möbius–Kantor graph is a symmetric bipartite cubic graph with 16 vertices and 24 edges named after August
Möbius–Kantor_graph
Abstract regular polyhedron with 10 triangular faces
K_{6}} (the complete graph with 6 vertices) on a real projective plane. With this embedding, the dual graph is the Petersen graph --- see hemi-dodecahedron
Hemi-icosahedron
Graph of numbers differing by a square
Paley graphs form an infinite family of conference graphs, which yield an infinite family of symmetric conference matrices. Paley graphs allow graph-theoretic
Paley_graph
Geometric space
g'<g} . Given a marked, stable, nodal curve one can associate its dual graph, a graph with vertices labelled by nonnegative integers and allowed to have
Moduli_of_algebraic_curves
Element of graph theory
{\displaystyle G} is a planar graph, and orientations of G {\displaystyle G} are transferred to orientations of the planar dual graph of G {\displaystyle G}
Acyclic_orientation
Mathematical graph theorem
this case, the dual graph is cubic and bridgeless, so by Petersen's theorem it has a matching, which corresponds in the original graph to a pairing of
Petersen's_theorem
Diagram of harmonic relations in music
second can be found halfway towards the major third. The Tonnetz is the dual graph of Schoenberg's chart of the regions, and of course vice versa. Research
Tonnetz
24-vertex symmetric bipartite cubic graph
Nauru graph is a toroidal graph: it consists of 12 hexagonal faces together with the 24 vertices and 36 edges of the Nauru graph. The dual graph of this
Nauru_graph
Unsolved problem in graph theory
The cut induced by such a partition in the dual graph corresponds to a Hamiltonian cycle in the primal graph. Although the truth of Barnette's conjecture
Barnette's_conjecture
Computation method in geometry
be also used to obtain a Voronoi diagram of the points, which is the dual graph of the Delaunay triangulation. The Bowyer–Watson algorithm is an incremental
Bowyer–Watson_algorithm
Catalan solid with 60 faces
icositetrahedron Conway, Symmetries of things, p.284-286 "Archimedean Dual Graph". Williams, Robert (1979). The Geometrical Foundation of Natural Structure:
Deltoidal_hexecontahedron
Framework in mathematics
DTW-equivalent shortest path problem to the maximum flow problem in the dual graph, which can be solved by most max-flow algorithms. However, when the data
Graphical_time_warping
{\mathcal {F}})\to J(X)\to J({\widetilde {X}})\to 0.} We next define the dual graph of X; a one-dimensional CW complex defined as follows. (related to whether
Complete_algebraic_curve
Bipartite, 3-regular undirected graph
field of graph theory, the Pappus graph is a bipartite, 3-regular, undirected graph with 18 vertices and 27 edges, formed as the Levi graph of the Pappus
Pappus_graph
Subdivision of the plane by lines
way of constructing the aperiodic Penrose tiling involves finding the dual graph of an arrangement of lines forming five parallel subsets. The maximum
Arrangement_of_lines
Abstraction of 2-colorable graphs
degree; Eulerian graphs may be disconnected. For planar graphs, the properties of being bipartite and Eulerian are dual: a planar graph is bipartite if
Bipartite_matroid
Bipartite non-Hamiltonian polyhedral 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
Herschel_graph
Polyhedron resembling a soccerball
represented as a polyhedral graph, meaning a planar graph (one that can be drawn without crossing edges) and 3-vertex-connected graph (remaining connected whenever
Truncated_icosahedron
Independence system partitionable into circuits
circuit. For planar graphs, the properties of being Eulerian and bipartite are dual: a planar graph is Eulerian if and only if its dual graph is bipartite.
Eulerian_matroid
Polyhedron with 9 faces
for convex polyhedra can only be quadrilaterals. An enumeration of the dual graphs of quadrilateral-faced polyhedra is given by Broersma, H. J.; Duijvestijn
Enneahedron
Type of plane partition
metrics. Voronoi diagrams of 20 points under two different metrics The dual graph for a Voronoi diagram (in the case of a Euclidean space with point sites)
Voronoi_diagram
Solid with eight equal triangular faces
octahedron give rise to a graph, a discrete structure drawn in a plane. The name is octahedral graph. The octahedral graph is an example of a four-connected
Regular_octahedron
Type of random graph
statistical mechanics, probability theory, graph theory, etc. the random cluster model is a random graph that generalizes and unifies the Ising model
Random_cluster_model
Algebraic encoding of graph connectivity
is a graph polynomial. It is a polynomial in two variables which plays an important role in graph theory. It is defined for every undirected graph G {\displaystyle
Tutte_polynomial
In the mathematical field of graph theory, the F26A graph is a symmetric bipartite cubic graph with 26 vertices and 39 edges. It has chromatic number 2
F26A_graph
Solid with 12 equal pentagonal faces
represented as a graph, and it is called the dodecahedral graph, a Platonic graph. This graph can also be constructed as the generalized Petersen graph G ( 10
Regular_dodecahedron
Solid with six equal square faces
drawing a graph with vertices connected with an edge in a plane. Such a graph is called the cubical graph, a special case of the hypercube graph. The cube
Cube
Partition of a graph's nodes into 2 disjoint subsets
In graph theory, a cut is a partition of the vertices of a graph into two disjoint subsets. Any cut determines a cut-set, the set of edges that have one
Cut_(graph_theory)
Archimedean solid with 62 faces
pentagrammic prisms. In the mathematical field of graph theory, a rhombicosidodecahedral graph is the graph of vertices and edges of the rhombicosidodecahedron
Rhombicosidodecahedron
Convex polyhedron projected from hypercube
map, and the 1-skeleton of the zonohedron can be viewed as the planar dual graph to an arrangement of great circles on the sphere. Conversely any arrangement
Zonohedron
On coloring the edges of graphs
In graph theory, Vizing's theorem states that every simple undirected graph may be edge colored using a number of colors that is at most one larger than
Vizing's_theorem
Graph containing cycles of all possible lengths
In the mathematical study of graph theory, a pancyclic graph is a directed graph or undirected graph that contains cycles of all possible lengths from
Pancyclic_graph
Matroid in which every permutation is a symmetry
n} -edge dipole graph, and the dual uniform matroid U n n − 1 {\displaystyle U{}_{n}^{n-1}} is the graphic matroid of its dual graph, the n {\displaystyle
Uniform_matroid
orientation of a spanning tree of the graph. The remaining edges, not in this tree, form a spanning tree of the dual graph, and their orientations can be chosen
Pfaffian_orientation
Graph which remains connected when fewer than k edges are removed
the graph. Edge connectivity is the dual concept to girth, the length of the shortest cycle in a graph, in the sense that the girth of a planar graph is
Edge_connectivity
Covering by shapes without overlaps or gaps
convex polygon. The Delaunay triangulation is a tessellation that is the dual graph of a Voronoi tessellation. Delaunay triangulations are useful in numerical
Tessellation
Undirected graph with 11 nodes and 27 edges
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 Anita
Goldner–Harary_graph
directed acyclic graph) in the sense that, if G is a planar graph, and orientations of G are transferred to orientations of the planar dual graph of G by turning
Strong_orientation
Convex polyhedron with 14 triangle faces
was incorrect. The Fritsch graph is one of only six graphs in which every neighborhood is a 4- or 5-vertex cycle. The dual polyhedron of the triaugmented
Triaugmented_triangular_prism
Games in game theory
particular game. Some players find it easier to color vertices of the dual graph, as in the Four color theorem. In this method of play, the regions are
Map-coloring_games
Discrete mathematics decomposition
representation, which is a rectangulation. The dual graph of a rectangulation is a triangulation of a 4-gon. On these graphs, we can put a transversal structure
Rectangulations
Graph describing a topological embedding
In topological graph theory, a graph-encoded map or gem is a method of encoding a cellular embedding of a graph using a different graph with four vertices
Graph-encoded_map
Geometry with 7 points and 7 lines
the Heawood graph (see figure). A bijection between the point set and the line set that preserves incidence is called a duality and a duality of order two
Fano_plane
Set of edges without common vertices
In the mathematical discipline of graph theory, a matching or independent edge set in an undirected graph is a set of edges without common vertices. In
Matching_(graph_theory)
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