(b)
K
3,3
is given below with partite sets U = {a, b, c} and
W = {A, B, C}.
Give a connected bipartite subgraph of K
3,3
with
the same partite sets but with U failing to be neighborly.
Explain why your example satisfies the requirements.
_________________________________________________________________
12. (15 pts.)
(a)
State Kuratowski’s Theorem that characterizes
planar graphs.
(b)
Without using Kuratowski’s Theorem, explain briefly how one
can see that K
5
is not planar.
(c)
Without using Kuratowski’s Theorem, explain briefly how one
can see that K
3,3
is not planar.
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_________________________________________________________________
13. (25 pts.)
(a)
(5 pts.)
What is a bridge?
[Hint: Original
definition please.]
(b)
(5 pts.)
What is a cutvertex??
[Definition, please.]
(c)
(15 pts.) For the graph G below, determine the cutvertices,
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):
MAD3305/Final Exam
Page 8 of 8
_________________________________________________________________
14. (25 pts.)
(a)
(5 pts.)
What is a network, N?? [Hint: It
would be nice to see Gould’s definition.]
(b)
(5 pts.)
What is a legal (or feasible) flow in a network
N ? [ Hint: Definition. ]
(c)
(15 pts.) Obtain a maximum flow f in the network below, and
verify the flow is a maximum by producing a set of vertices S
that produces a minimum cut. Check that the total capacity of
that cut is the same as the value of your max flow.
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 Summer '12
 Rittered
 Graph Theory, Planar graph, 5 pts, outdegree, D. König

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