Mathematics for Computer Science

“mcs” — 2017/3/3 — 11:21 — page 508 — #516
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Chapter 12
Simple Graphs
Problem 12.46. (a) How many spanning trees are there for the graph G in Figure
12.33?
(b) For G e, the graph G with vertex e deleted, describe two spanning trees that
have no edges in common.
(c) For G e with edge ha—di deleted, explain why there cannot be two edgedisjoint
spanning trees.
Hint: : Count vertices and edges.
Problem 12.47.
Prove that if G is a forest and
then G is a tree.
jV .G/j D jE.G/j C 1; (12.5)
Problem 12.48.
Let H 3 be the graph shown in Figure 12.34. Explain why it is impossible to find
two spanning trees of H 3 that have no edges in common.
000
010
100
110
101
111
001
011
Figure 12.34 H 3 .
Exam Problems
Problem 12.49. (a) Let T be a tree and e a new edge between two vertices of T .
Explain why T C e must contain a cycle.
(b) Conclude that T C e must have another spanning tree besides T .