Question

$1$

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$1$

$4$

$$marks

$)$

$$Let

$$

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$($

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$$

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$$be a connected, undirected graph.

a

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$[$

$7$

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$$Assume that

$$

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$,$

$$

$2$

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$2$

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$$and

$$

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$3$

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$$are spanning

trees of

$$

$$and that their sets of edges meet the following conditions:

$$

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$\backslash$cap

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$2$

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$$

$,$

$$

$1$

$\backslash$cap

$$

$3$

$=$

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$$and

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$$

$3$

$=$

$$

$.$

$($

$$

$$represents the empty set.

$)$

What is the smallest number of vertices that G can have? Explain your answer

and provide an example of such a graph. Clearly describe the

$3$

$$spanning trees in

the example you provide.

b

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$7$

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$$Let

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$$be a graph. Provide an algorithmic strategy that can find

and output exactly two spanning trees of

$$

$$that do not share any edges, e

$.$

g

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$,$

$$

$1$

$=$

$($

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$$

$1$

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$,$

$$

$2$

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$$

$2$

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$$where

$$

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$$and

$$

$2$

$$are spanning trees of

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$$and

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$$

$2$

$=$

$$

$,$

$$if such two spanning trees exist.

Provide your algorithmic strategy using pseudo code or a detailed description in

natural language. Explain why your algorithmic strategy meets the requirement.