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Section 12.4
Exercises 12.4
1. (a) [BB] This is acyclic:
F, B, D, H, A, E,
C, G is a canonical labeling.
(b) This has cycle
DH F AED.
(c) This is acyclic: C,
H, I, F, B,
G,
D, E, A
is a canonical labeling.
(d) This digraph has cycle
C
EF DC.
343
2. indeg
Vo
=
0 because there are no arcs
ViVO
with
i
>
O. There are no restrictions on the outdegree of
Vo,
which can be any integer between 0 and
n
 1 (inclusive).
3. [BB] Let
9
be a digraph with
n
vertices and let
A
be the adjacency matrix of
g.
The indegree of vertex
i is the sum of the entries in column i. This requires
n

1 twonumber additions. Repeating for
n
vertices involves
n(n

1) =
O(n
2
)
additions.
4. (a) We use the algorithm described in Theorem 12.4.3.
Given an acyclic digraph with vertex set
V
=
{O, 1,2,
...
,n

I} to find a canonical labeling,
Step 1: Let
V
=
{O, 1,
... ,
n
 I} be the vertex set and let
t
=
O.
Step 2: While
t
~
n
 1
• Let
Vt
be a vertex in
V
of indegree O.
• Replace
V
by
V
'
{Vt}
and compute the indegrees of the vertices of (this new)
V.
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 Summer '10
 any
 Graph Theory

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