Basic network with a contextual link

This was prepared to demonstrate the network drawn from one of the practice assessments.
David delivers supplies every week to shops along each road and in each of the towns. Can he traverse the network of roads without repeating any roads? Explain your answer. At present David lives in Hamilton and once a month visits his mother in Wairoa. What is the shortest distance he has to travel to visit his mother? You may use Djikstra method or otherwise to explain the process. Occasionally David delivers only to the shops actually in the towns. Describe a possible route that visits each town just once and minimises the distance David will have to travel. Give both the route and the minimum distance he will have to travel.