Dual graph
Let me jump ahead here and tell you a little bit about how a polyhedron and its dual are related. A polyhedron P and its dual P* have the same number of edges,..
Jun 28, 2011.. Endow the rational numbers (or any global field) with the discrete topology, what will be the (compact) Pontryagin dual of the additive group..
In the mathematical discipline of graph theory, the dual graph of a plane graph G is a graph that has a vertex corresponding to each face of G, and an edge joining two neighboring faces for each edge in G. The term "dual" is used because this property is symmetric, meaning that if H is a dual of G, then G is a dual of H (if G is connected). The same notion of duality may also be used for more general embeddings of graphs in manifolds. The notion described in this page is different from the edge-to-vertex dual (Line graph) of a graph and should not be confused with it. The dual of a plane graph is a plane multigraph - multiple edges. If G is a connected plane graph and if G′ is the dual of G then G is isomorphic to the dual of G′. Since the dual graph depends on a particular embedding, the dual graph of a planar graph is not unique in the sense that the same planar graph can have non-isomorphic dual graphs. In the picture, the red graphs are not isomorphic because the upper one has a vertex with degree 6 (the outer face). However, if the graph is 3-connected, then Whitney showed that the embedding, and thus the dual graph, is unique. Because of the duality, any result involving counting faces and vertices can be dualized by exchanging them. Let G be a connected graph. An algebraic dual of G is a graph G★ such that G and G★ have the same set of edges, any cycle of G is a cut of G★, and any cut of G is a cycle of G★. Every planar graph has an algebraic dual, which is in general not unique (any dual defined by a plane embedding will do). The converse is actually true, as settled by Hassler Whitney in the Whitney's planarity criterion: A connected graph G is planar if and only if it has an algebraic dual. The same fact can be expressed in the theory of matroids: if M is the graphic matroid of a graph G, then the dual matroid of M is a graphic matroid if and only if G is planar. If G is planar, the dual matroid is the graphic matroid of the dual graph of G. The weak dual of a plane graph is the subgraph of the dual graph whose vertices correspond to the bounded faces of the primal graph. A plane graph is outerplanar if and only if its weak dual is a forest, and a plane graph is a Halin graph if and only if its weak dual is biconnected and outerplanar. For any plane graph G, let G+ be the plane multigraph formed by adding a single new vertex v in the unbounded face of G, and connecting v to each vertex of the outer face (multiple times, if a vertex appears multiple times on the boundary of the outer face); then, G is the weak dual of the (plane) dual of G+.
Oct 11, 2002.. In particular, we propose that the singlet sector of the well-known critical 3-d O(N) model with the $(\phi^a \phi^a)^2$ interaction is dual, in the..
Feb 14, 2015.. The Hodge operator maps. Using the Hodge dual we can dualize some operations. For example. where is the codifferential. If is a vector field..
The dual of a Platonic solid or Archimedean solid can be computed by connecting the midpoints of the sides surrounding each polyhedron vertex (the vertex..
Duality. For every polyhedron there exists a dual polyhedron. Starting with any regular polyhedron, the dual can be constructed in the following manner: (1)..
A concept defined for a partial order P will correspond to a dual concept on the.. element of Pd: minimality and maximality are dual concepts in order theory.
Nov 8, 2011.. We also present somewhat weaker evidence that the 3-state and tricritical Potts models are dual to pure higher spin theories of gravity based..
By the duality principle, for every polyhedron, there exists another polyhedron in which faces and polyhedron vertices occupy complementary locations. This polyhedron is known as the dual, or reciprocal. The process of taking the dual is also called reciprocation, or polar reciprocation. Brückner (1900) was among the first to give a precise definition of duality (Wenninger 1983...