5 [Answer this question on the insert provided.]
\includegraphics[max width=\textwidth, alt={}, center]{b1227633-913e-41a9-8bf8-1f064056963e-3_659_1002_324_609} In this network the vertices represent towns, the arcs represent roads and the weights on the arcs show the shortest distances in kilometres.

1. The diagram on the insert shows the result of deleting vertex \(F\) and all the arcs joined to \(F\). Show that a lower bound for the length of the travelling salesperson problem on the original network is 38 km . The corresponding lower bounds by deleting each of the other vertices are: $$A : 40 \mathrm {~km} , \quad B : 39 \mathrm {~km} , \quad C : 35 \mathrm {~km} , \quad D : 37 \mathrm {~km} , \quad E : 35 \mathrm {~km} \text {. }$$ The route \(A - B - C - D - E - F - A\) has length 47 km .
2. Using only this information, what are the best upper and lower bounds for the length of the solution to the travelling salesperson problem on the network?
3. By considering the orders in which vertices \(C , D\) and \(E\) can be visited, find the best upper bound given by a route of the form \(A - B - \ldots - F - A\).