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The construction of a dual D(G) can be applied in any plane graph G: draw a...

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Advance Math

The construction of a dual D(G) can be applied in any planegraph G: draw a vertex of D(G) in the middle of each region of Gand draw an edge e* of D(G) perpendicular to each edge e of G; e*connects the vertices of D(G) representing the regions on eitherside of e.

a) A dual need not be a graph. It might have two edges betweenthe same pair of vertices or a self-loop edge (from a vertex toitself). find two planar graphs with duals that are not graphsbecause they contain these two forbidden situations.

C) Show that the degree of a vertex in dual graph D(G) equalsthe number of boundary edges of the corresponding region in theplanar graph G.

E) show for any plane depiction of a graph G that the verticesof G correspond to regions in D(G)

Answer & Explanation Solved by verified expert
4.1 Ratings (668 Votes)
aC In the example we have drawndegree w1 3 the region whose middle point is w1 hasthree boundary edges v1 v4 v1 v2 v2 v4degree w2 3    See Answer
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