Optimization, Networks and Graphs

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a) Give an example, with justification, of a graph G which is:
(i) bipartite and has a perfect matching,
(ii) bipartite and has no perfect matchings,
[5 marks]
Let G = (V, E) be a bipartite graph with parts V1 and V2 and suppose that G
has a perfect matching.
(b) Show that |V1| = |V2|.
[5 marks]
(c) Let G denote the complement of G. Show that the clique number ω(G)
satisfies
ω(G) ≥ |V1|.
[5 marks]
(d) Let W ⊆ V be any clique in G and let W1 := W ∩V1 and W2 := W ∩V2.
Stating clearly any results that you use, prove that
|W | = |W1| + |W2| ≤ |V2|.
[Hint: You might want to recall Hall’s Marriage Theorem.]
[5 marks]
(e) Prove that there exists a colouring of G with |V1| colours.
[5 marks]
(f) Deduce that
ω(G) = χ(G) = |V1|.

Sample Solution

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Diversity in Media

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The popular cultural example you must use is either the movie “The Blind Side” or TV show “Grey’s anatomy”
ONLY select one
-Analyze the representation and social message that is associated with the example
-What is the composition of the primary characters, and what groups within diversity do they represent?
-What is the latent (or unintended) social message or commentary about a particular group to society that is
reflected by this example?
-What are the social implications of this message? Briefly discuss any potential social impact or response to
the example.
-use specifics

Sample Solution

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Examination of global issue

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Discuss the global issue you chose by briefly examining that issue as it exists in the broad picture not just as it exists in the US.
Discuss why that issue affects you the most personally. Offer us your evidence and rationale for choosing this particular issues
Then tell us the steps you can take to prevent this issue from becoming a problem in the US.

Sample Solution

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Vulnerability Identification

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The first step in a pen test is to gather as much information you can on your target company. This consists
of how much you can find out about their company concerning their overall network, personnel, technology.
This may be infrastructure or desktops, including hardware and software, e-mail addresses, etc. Record all
your information in Chapter 3, Information Gathering Risk section, indicating what you were able to find, how
you found it, and what you may recommend to the (company) as a way to limit the amount of information that
can be found about them. Also, include any names and associated e-mail addresses you find from
management that could be used later in a social engineering attack. The only rule for information gathering is
that you are NOT allowed to touch the company’s network. At all. Use single third-party resources. This means
you are NOT to use NMAP or any other type of scanner to scan the existing network.

Sample Solution

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