The picture below represents the social network of a karate

club that you have been studying over the past year. Unfortunately, there is an argument between the administrator $($node $1)$ and the coach

$($node $9).$ As a result the coach has started her own club and asked the students to join

her. You expect that

$1.$ All of the coach$$s friends will join the new club, no matter what.

$2.$ Initially, everyone else will initially stay in the current club.

$3.$ Then, at each time step every other person will choose the club in which he or she

has the highest number of friends. When a person has an equal number of friends in

both clubs, he or she may choose a club arbitrarily.

Considering all this information:

$1.$ Give two different ways the club can break into two groups.

$2.$ Now, consider what would happen if the administrator $($node $1)$broke off$$ instead

of the coach $($node $9)$breaking off$.$ This would mean that all of the administrator$$s

friends would join the new club, and everyone else would initially stay in the old club.

In this scenario, give two different ways the club can break into two groups.

$3.$ Let the coach be the one breaking off. The coach approaches you and says she can

try to befriend one more person in the group before breaking off. That person may

not be one of the direct friends of the administrator, but she can try anyone else.

Who should she try to befriend to increase the size of her club the most?Unfortunately, there is an argument between the administrator $($node $1)$ and the coach

$($node $9).$ As a result the coach has started her own club and asked the students to join

her. You expect that

$1.$ All of the coach$$s friends will join the new club, no matter what.

$2.$ Initially, everyone else will initially stay in the current club.

$3.$ Then, at each time step every other person will choose the club in which he or she

has the highest number of friends. When a person has an equal number of friends in

both clubs, he or she may choose a club arbitrarily.

Considering all this information:

$1.$ Give two different ways the club can break into two groups.

$2.$ Now, consider what would happen if the administrator $($node $1)$broke off$$ instead

of the coach $($node $9)$breaking off$.$ This would mean that all of the administrator$$s

friends would join the new club, and everyone else would initially stay in the old club.

In this scenario, give two different ways the club can break into two groups.

$3.$ Let the coach be the one breaking off. The coach approaches you and says she can

try to befriend one more person in the group before breaking off. That person may

not be one of the direct friends of the administrator, but she can try anyone else.

Who should she try to befriend to increase the size of her club the most?Unfortunately, there is an argument between the administrator $($node $1)$ and the coach

$($node $9).$ As a result the coach has started her own club and asked the students to join

her. You expect that

All of the coach's friends will join the new club, no matter what.

Initially, everyone else will initially stay in the current club.

Then, at each time step every other person will choose the club in which he or she

has the highest number of friends. When a person has an equal number of friends in

both clubs, he or she may choose a club arbitrarily.

Considering all this information:

Give two different ways the club can break into two groups.

Now, consider what would happen if the administrator $($node $1)$ "broke off" instead

of the coach $($node $9)$ "breaking off". This would mean that all of the administrator's

friends would join the new club, and everyone else would initially stay in the old club.

In this scenario, give two different ways the club can break into two groups.

Let the coach be the one breaking off. The coach approaches you and says she can

try to befriend one more person in the group before breaking off. That person may

not be one of the direct friends of the administrator, but she can try anyone else.

Who should she try to befriend to increase the size of her club the most?