# dual graph

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## dual graph

[¦dü·əl ′graf]
(mathematics)
A planar graph corresponding to a planar map obtained by replacing each country with its capital and each common boundary by an arc joining the two countries.
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Then E is a triple fork and F is a quadruple fork in dual graph.
is no longer log canonical since the dual graph of F is a quadruple fork which is not in the classification of dual graph of log canonical singularities in [KM98, Theorem 4.
For instance, to study curves on a general K3 surface, we can let it degenerate to a union of projective planes, the dual graph of which is a triangulation of the real 2-sphere.
Every plane graph has a dual graph , formed by assigning a vertex of , to each face of and joining two vertices of by edges if and only if the corresponding faces of share edges in their boundaries.
For each stepped surface [partial derivative]U, the associated graph [GAMMA](U) is obtained by starting with the planar dual graph and replacing each vertex by a small quadrilateral.
Consider the dual graph D of T, which contains one vertex for each region that is bounded by edges of T and the boundary of the convex hull (excluding the region outside the convex hull of G).
x,y] be the geometric dual graph of the plane graph [G.
The four regions 00, 01, 10 and 11 in that figure are the nodes of the dual graph [ILLUSTRATION FOR FIGURE 1 OMITTED] [C.
Now observe the graphs in Figure 2 for n = 2, 3, and 4, which correspond to reduced (because there is no loop or bridge), alternating (because the labeling is either all R or L)achiral (because the dual graph is isomorphic and so isequivalent).
Next we construct a dual graph of D by converting the dark regions in D into vertices in a graph G and converting the crossings in D between two dark regions into edges incident to the corresponding vertices in G.
The facets 12 and 13 with the vertices {0, 2, 5, 7, 8} and {1, 2, 5, 7, 8}, respectively, are adjacent in the dual graph (via the common ridge with vertex set {2, 5, 7,8}).
Keywords: Reidemeister moves, Dual graphs, ambient isotopy invariants, reduced alternating knots , equivalent knots .
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