Choosing & Justifying Methods — practice questions

For each situation, name which of the three network methods applies and give a one-sentence reason: (i) a power company connecting six new subdivisions to the grid for the least cable; (ii) a driver finding the quickest way from the airport to the stadium; (iii) a council checking every stretch of footpath in a park.

A network of five sites has arcs $AB = 6$, $AC = 3$, $BC = 4$, $BD = 7$, $CD = 5$, $CE = 9$, $DE = 2$. Find the minimum spanning tree and the shortest path from A to E, and explain why the two answers use different arcs.

A council will build a fibre network linking six sites using the minimum spanning tree ($AB=4$, $AC=6$, $BC=3$, $BD=8$, $CD=5$, $CE=11$, $DE=4$, $DF=9$, $EF=3$; MST total 19 km). The engineers argue that a tree is too fragile and want the network to survive any single cable break. Determine the cheapest network that meets that requirement, and evaluate whether the council should pay for it.