or your long term strategic plan, you would like to identify where to place offices for your business. Although you would like to have many offices, it is essential that each pair can co mmunicaticate with each other in a cost effective manner. using your knowledge, you choose to represent the information as a graph, where potential office locations are represented by vertices and the cost of communication between a pair of locations is represented by a positive weight on an edge.

we use the term cluster to refer to a set of vertices $($Represented by their IDs$)$ such that there is an edge between each pair of vertices in the set. The total value of$.$ a cluster is the sum of the weights of the edges with both end points in the cluster. For example, in sample graph $4,\{$e$\}$ is a cluster of size $1($with total value $0,$ since there are no edges with end points in the cluster$),\{$e$,$f$\}$ is a cluster of size $2($with total value equal to $3,$ the weight of the edge ef$),$ and $\{$e$,$f$,$g$\}$ is a cluster of size $3($with total value equal to the sum of the weights of edges ef$,$fg and eg or $3+4+9=16),$ The set $\{$e$,$g$,$d$\}$ is not. a cluster, since there is no edge ed$.$

please see the reference page on the module graphs.py for illustrations of the sample graphs.

a$)$ in sample graph $4,$ what is the max size of a cluster and which cluster of max size has the minimum total value? if there is more than one correct answer any will do$.$ briefly justofy your answer explaining why it is a cluster. why its size is the max possible and why its total value is the min possible. i have attached it

b$)$ give the input and output specifications for an enumeration problem for the problem of finding clusters of size k with the min total value. you may use the terms defined above without supplying definitions.

c$)$ by adding a bound on the total value, give the input and output specifications for a related decision problem.

d$)$ give a different example of a real work situation to which the decision problem can be applied, this time when the edge weights are all equal to $1.$ your situation does not have to be useful. even ridiculous situations are acceptable, a long as you justify why the problem applies.