californiarepoman
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- Jul 24, 2013
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a. Find a dual graph for each of the two planar graphs and the one planar multigraph in Fig. 11.72
b. Does the dual for the multigraph in part (c) have any pendant vertices? If not, does this contradict the third observation made prior to Definition 11.20?
The first graph "G" has 3 faces: a bounded face inside the circuit adhe, another inside "acgb", and the unbounded face. It's dual "G" will have a vertex for each face "uvw". The 2 bounded faces have no edge in common, so in "G" there is no edge between "uv". Each bounded face had 4 edges in common with the unbounded face so there must be 4 edges between "uw" and 4 more between "vw". The edge ef has the unbounded face on both sides so "G" has a lop at "W".