Mathematics for Computer Science

“mcs” — 2017/3/3 — 11:21 — page 540 — #548
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Chapter 13
Planar Graphs
(c) A connected simple graph has one more vertex than it has edges. Explain why
it is a planar graph.
(d) How many faces does a planar graph from part c have?
(e) How many distinct isomorphisms are there between the graph given in Figure
13.17 and itself? (Include the identity isomorphism.)
a
d e b f
c
Figure 13.17
Class Problems
x
Problem 13.6.
Figure 13.18 shows four different pictures of planar graphs.
(a) For each picture, describe its discrete faces (closed walks that define the region
borders).
(b) Which of the pictured graphs are isomorphic? Which pictures represent the
same planar embedding?—that is, they have the same discrete faces.
(c) Describe a way to construct the embedding in Figure 4 according to the recursive
Definition 13.2.2 of planar embedding. For each application of a constructor
rule, be sure to indicate the faces (cycles) to which the rule was applied and the
cycles which result from the application.
Problem 13.7.
Prove the following assertions by structural induction on the definition of planar
embedding.
(a) In a planar embedding of a graph, each edge occurs exactly twice in the faces
of the embedding.