In a given locality, it is intended that 5 villages (A1, A2, A3, A4 and A5) be connected to each other and that the cost of this connection be as low as possible. The graph below shows the costs of the connections that can be made between the different villages:

Given the information provided,

a) comment, justifying conveniently, the following statement "Once a village is chosen to start a race, it is possible to pass through all the villages and return to the starting village using the route corresponding to one of the connection schemes with minimum cost".

b) indicate the minimum cost of the route that allows connecting all the villages and a graphical representation of the corresponding graph. Justify the choice of the method used and present all the steps performed, with the corresponding justifications.

A4 75 A3 16 72 A1 31 67 23 62 104 A2 54 A5