The red graph is the dual graph of the blue graph, and vice versa.
In the 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 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 (graphs that are already embedded in the plane) rather than planar graphs (graphs that may be embedded but for which the embedding is not yet known). For planar graphs generally, there may be multiple dual graphs, depending on the choice of planar embedding of the graph.
Historically, the first form of graph duality to be recognized was the association of the Platonic solids into pairs of dual polyhedra. Graph duality is a topological generalization of the geometric concepts of dual polyhedra and dual tessellations, and is in turn generalized combinatorially by the concept of a dual matroid. Variations of planar graph duality include a version of duality for directed graphs, and duality for graphs embedded onto non-planar two-dimensional surfaces.
These notions of dual graphs should not be confused with a different notion, the edge-to-vertex dual or line graph of a graph.
The term dual is used because the property of being a dual graph is symmetric, meaning that if H is a dual of a connected graph G, then G is a dual of H. When discussing the dual of a graph G, the graph G itself may be referred to as the "primal graph". Many other graph properties and structures may be translated into other natural properties and structures of the dual. For instance, cycles are dual to cuts, spanning trees are dual to the complements of spanning trees, and simple graphs (without parallel edges or self-loops) are dual to 3-edge-connected graphs.
Graph duality can help explain the structure of mazes and of drainage basins. Dual graphs have also been applied in computer vision, computational geometry, mesh generation, and the design of integrated circuits.
mathematical discipline of graph theory, the dualgraph of a planar graph G is a graph that has a vertex for each face of G. The dualgraph has an edge for each...
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...
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...
form a graph embedded on this surface, and the vertices and edges of the (abstract) dual polyhedron form the dualgraph of the original graph. An abstract...
from the similarly-named notions of a dual polyhedron, and of the dualgraph of a surface-embedded graph. Dualgraphs have also been used to define and study...
useful graph that is often associated with a triangulation of a polygon P is the dualgraph. Given a triangulation TP of P, one defines the graph G(TP)...
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...
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...
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...
planar graphs, and have a unique planar embedding. More specifically, every wheel graph is a Halin graph. They are self-dual: the planar dual of any wheel...
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...
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...
of graph; dualgraph; medial graph; quotient graph; Y-Δ transform; Mycielskian. Binary operations create a new graph from two initial graphs G1 = (V1,...
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...
In mathematics, random graph is the general term to refer to probability distributions over graphs. Random graphs may be described simply by a probability...
graph. In this case, the dual of M {\displaystyle M} is the matroid of the dualgraph of G . {\displaystyle G~.} The dual of a vector matroid representable...
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...
The dualgraph of a directed graph, embedded in the plane, is a graph with a vertex for each face of the given graph, and a dual edge between two dual vertices...
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...
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 dualgraph. A spanning...
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