9.01 Minimum connector problems

The council wants to lay down new electric cables between the houses in a particular area. The graph displays the map of the area:

To minimise the cost of cables, the council wants to connect the houses in such a way that they minimise the cable length. Construct a graph that would achieve this.

Water pipes need to be laid down in a new area that is undergoing construction. The given pipe map was suggested:

To minimise the cost, the council wants to connect every house to water, minimising the total length of water pipes used. Construct a graph that would achieve this.

The following weighted network shows possible connections (and associated costs, in millions of dollars) for connecting rural towns to a common cabled internet service.

To minimise the cost of connecting each town to the network, we want to find the minimal spanning tree.

The direct flight routes between 5 major cities of the same country are shown in the given graph. Each edge is marked with the distance (in \text{km}) travelled on each route:

To optimise the investment put into these routes, the operating airline wants to find the minimum spanning tree for the graph.

At a university, there is a new plan to connect all the campus buildings using underground fiber optic cables. The following network was suggested by the engineer:

The project accountant wants to minimise the cost of the project. He realises that there are unnecessary cables that can be removed to minimise cable length.

Construct the network that represents the least cost.

In a certain area, new underground cables need to be laid to connect electric substations. The following graph displays the current connection layout:

Since the project has a limited budget, the engineers have to design a connection layout that uses the minimum cable length.

Construct the graph that represents the connection layout with the minimum amount of cable.

An airplane company wants to redesign their flight routes to make it cost efficient. The following map displays the current flight routes:

In order for the flight routes to be efficient, each city needs to be connected and the sum of the route distances needs to be minimised.

Construct the graph that will achieve this.

The bus company decided to redesign its travel routes after some new roads were built. The following map shows the routes connecting the towns and the times (in minutes) taken to cover the route:

The bus company wants to connect all towns with the most time efficient route.

Construct the graph that will achieve this.

A large amusement park is trying to implement a new security system such that all security posts are connected by electric cabling. The following map shows all possible connections and their lengths:

Since the project has a limited budget, the engineers have to minimise the length of cabling used.

Construct the graph that represents the connections using the least amount of cable.

Urban planners are trying to connect houses in a certain area.

The following map represents all possible routes between houses in the area, and the distances between them:

The urban planners wish to create roads that are most distance efficient, and allow them to connect each house.

Construct the graph that is the most distance efficient.

The water authority wants to lay water pipes along the roads in order to put a fire hydrant at every node on the network shown on the following map. The numbers represent the length of the pipe that connects any two nodes:

To minimise the cost of the operation, the water authority wants to minimise the length of the pipes.

Construct the graph that will achieve this.

The rangers of a national park want to clear walking tracks to 6 landmarks in the area. The potential tracks between landmarks are shown in the given graph, along with the number of days required to clear each track for use:

To make a network of tracks in the shortest time, the rangers want to find a minimal spanning tree for the graph.