7 A tour guide wants to find a route around eight places of interest: Queen Elizabeth Olympic Park ( \(Q\) ), Royal Albert Hall ( \(R\) ), Statue of Eros ( \(S\) ), Tower Bridge ( \(T\) ), Westminster Abbey ( \(W\) ), St Paul's Cathedral ( \(X\) ), York House ( \(Y\) ) and Museum of Zoology ( \(Z\) ). The table below shows the travel times, in minutes, from each of the eight places to each of the other places.

1. Use the nearest neighbour method to find an upper bound for the minimum time to travel to each of the eight places, starting and finishing at \(Y\). Write down the route and give the time in minutes.
2. The Answer Book lists the arcs by increasing order of weight (reading across the rows). Apply Kruskal's algorithm to this list to find the minimum spanning tree for all eight places. Draw your tree and give its total weight.
3. (a) Vertex \(Q\) and all arcs joined to \(Q\) are temporarily removed. Use your answer to part (ii) to write down the weight of the minimum spanning tree for the seven vertices \(R , S , T , W , X , Y\) and \(Z\).
   (b) Use your answer to part (iii)(a) to find a lower bound for the minimum time to travel to each of the eight places of interest, starting and finishing at \(Y\). The tour guide allows for a 5 -minute stop at each of \(S\) and \(Y\), a 10 -minute stop at \(T\) and a 30 -minute stop at each of \(Q , R , W , X\) and \(Z\). The tour guide wants to find a route, starting and ending at \(Y\), in which the tour (including the stops) can be completed in five hours (300 minutes).
4. Use the nearest neighbour method, starting at \(Q\), to find a closed route through each vertex. Hence find a route for the tour, showing that it can be completed in time.