bridges, and blocks of G. List the cutvertices and bridges in
the appropriate places, and provide carefully labelled sketches
of the blocks.
Cutvertices:
Bridge(s):
Block(s):
_________________________________________________________________
3. (10 pts.)
Below, provide a proof by induction on the order
of the graph G that every nontrivial connected graph G has a
spanning tree.
[Hint: If the order of the graph G is at least
3, Theorem 1.10 implies that G has a vertex v with G  v
connected.]
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TEST2/MAD3305 Page 3 of 4
_________________________________________________________________
5. (10 pts.)
Apply Kruskal’s algorithm to find a minimum
spanning tree in the weighted graph below. 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?
_________________________________________________________________
6. (15 pts.)
(a) Suppose G
1
and G
2
are nontrivial graphs.
What does it mean mathematically to say that G
1
and G
2
are
isomorphic?? [This is really a request for the definition!]
(b) Sketch two graphs G and H that have the degree sequence
s: 2, 2, 2, 2, 2, 2 and have the same order and size, but are not
isomorphic. Explain briefly how one can readily see that the
graphs are not isomorphic.
(c) Explicitly realize C
5
and its complement below. [You may
provide carefully labelled sketches.] Next, explicitly define an
isomorphism from C
5
to its complement that reveals that C
5
is
selfcomplementary.
TEST2/MAD3305 Page 4 of 4
_________________________________________________________________
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?
_________________________________________________________________
8. (15 pts.)
(a) If G is a nontrivial graph, how is
κ
(G), the
vertex connectivity of G, defined?
(b) If G is a nontrivial graph, it is not true generally that if
v is an arbitrary vertex of G, then either
κ
(G  v) =
κ
(G)  1 or
κ
(G  v) =
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.]
(c) Despite the example above, if G is a nontrivial graph and v
is a vertex of G,
κ
(G  v)
≥
κ
(G)  1. Provide the simple proof
for this.
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 Summer '12
 Rittered
 Graph Theory, Vertex, 10 pts, κ, 15 pts, 25 pts

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