TEST2/MAD3305
Page 4 of 4
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7. (10 pts.)
Find a minimum spanning tree for the weighted
graph below by using only Prim’s algorithm and starting with the
vertex g.
When you do this, list the edges in the order that you
select them from left to right.
What is the weight w(T) of your
minimum spanning tree T?
Beginning with
vertex g, you should
obtain edges in the
following order:
ge, ef, fb, ba, ed,
dc.
There is only one
spanning tree, and
its weight is
w(T) = 31.
You can check the
edges, but not their
order using
Kruskal’s algorithm.
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8. (15 pts.)
(a)
If G is a nontrivial graph, how is
κ
(G), the
vertex connectivity of G, defined?
If G is a complete graph of order n, then
κ
(
G
)=n1
.
Otherwise, G has a vertexcut.
In this case,
κ
(G) = k where k is
the cardinality of a minimum vertexcut.
(b)
If G is a nontrivial graph, it is not true generally that if
v is an arbitrary vertex of G, then either
κ
(
Gv
)=
κ
(
G
)1o
r
κ
(
Gv
)=
κ
(G).
Give a simple example of a connected graph G
illustrating this.
[A carefully labelled drawing with a brief
explanation will provide an appropriate answer.]
Evidently, G
≅
K
1
+(
K
1
∪
K
2
).
Since G is
connected and a is a cutvertex of G,
κ
(G) = 1.
Note, however,Gb
≅
K
3
, and
thus
κ
(
Gb
)=2
,n
o
t0o
r1
.
(c)
Despite the example above, if G is a
nontrivial graph and v is a vertex of G,
κ
(
Gv
)
≥κ
(G)  1.
Provide the simple
proof for this.
Proof:
Let v be an arbitrary vertex of G.
If G is a complete
graph of order k, thenGvi
s complete of orderk1
.
Thus,
the conclusion follows from definition of
κ
.
So suppose G is not
a complete graph.
ThenGvi
sn
o
ta complete graph, too.
Then either
κ
(
Gv
)
≥κ
(
G
)1o
r
κ
(
Gv
)<
κ
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 Summer '12
 Rittered
 Graph Theory, Vertex, Planar graph, κ

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