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
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 Graph Theory, Polyhedron, Directed acyclic graph, canonical labeling

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