What is a dual in graph theory?
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 separated from each other by an edge, and a self-loop when the same face appears on both sides of an edge.
What is lattice in graph?
A lattice graph, also known as a mesh graph or grid graph, is a graph possessing an embedding in a Euclidean space. that forms a regular tiling. Examples include grid graphs and triangular grid graphs. Rook graphs are sometimes also known as lattice graphs (e.g., Brouwer).
Is the dual of a graph always connected?
If we follow the line from xF to xE, we “describe a path” in the dual graph from F to the external face. Thus, each vertex of the dual graph is connected to the vertex corresponding to the external face, which means that the dual graph must be connected.
Which of the following graph is self dual graph?
Wheel graphs are self-dual, as are the examples illustrated above. Naturally, the skeleton of a self-dual polyhedron is a self-dual graph. Since the skeleton of a pyramid is a wheel graph, it follows that pyramids are also self-dual.
What is the necessary and sufficient condition for two graphs to be duals of each other?
THEOREM. Prove that a neccessary and sufficient con- dition for two planar graphs G1 and G2 to be dual of each other is that there is one-one correspondence between the edges in G1 and edges in G2 such that a set of edges in G1 forms a circuit iff the corresponding set in G2 forms a cut-set.
What is a trellis bar graph?
Trellis plots, Trellis charts or Trellis graphs (also sometimes termed “Trellis displays” or simply “Trellis”) are a means of graphing multivariate data, as an array of M X N panels.
What is lattice R?
lattice: Trellis Graphics for R A powerful and elegant high-level data visualization system inspired by Trellis graphics, with an emphasis on multivariate data. Lattice is sufficient for typical graphics needs, and is also flexible enough to handle most nonstandard requirements.
Is the dual of a planar graph planar?
The planar dual of an embedded planar graph G is the graph G formed by placing a vertex inside each face of G, and connecting those vertices of G whose corresponding faces in G share an edge.
What are kuratowski’s two graphs?
A Kuratowski graph of the first type consists of the edges of a tetrahedron and one other segment joining the midpoints of two non-intersecting edges. A Kuratowski graph of the second type is the complete graph spanned by the vertices of a tetrahedron and a point in its interior. A graph G is planar (cf.
What is trellis display?
Trellis display is a framework for the visualization of data. On each panel of the trellis, a subset of the data is graphed by a display method such as a scatterplot, curve plot, boxplot, 3-D wireframe, normal quantile plot, or dot plot.
Which is the first form of graph duality?
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 algebraically by the concept of a dual matroid.
Why is the dual of a plane graph bidirectional?
Because the dual of the dual of a connected plane graph is isomorphic to the primal graph, each of these pairings is bidirectional: if concept X in a planar graph corresponds to concept Y in the dual graph, then concept Y in a planar graph corresponds to concept X in the dual.
How are dual graphs used in computer vision?
Dual graphs have also been applied in computer vision, computational geometry, mesh generation, and the design of integrated circuits . The unique planar embedding of a cycle graph divides the plane into only two regions, the inside and outside of the cycle, by the Jordan curve theorem.
When is the embedding of a dual graph isomorphic?
For some planar graphs that are not 3-vertex-connected, such as the complete bipartite graph K2,4, the embedding is not unique, but all embeddings are isomorphic. When this happens, correspondingly, all dual graphs are isomorphic.