Chinese Postman and Euler Tour Problems in Bi-Directed Graphs

Abstract

Euler tours occupy an interesting position in the history of graph theory. Current interest in this area is due to problems involving tours where service is required along arcs of the tour rather than at nodes. Examples of problems of this type involve mail delivery, snow removal, street cleaning, trash pickup, etc. In considering such problems involving city streets, the nodes are intersections and the arcs are roadways between intersections. However, there are one-way streets and two-way streets. If the service must be done on both sides of a two-way street then two one-way streets (one in each direction) can replace it. However, there are situations where the street, even though two-way, need only be traversed once. For example, mail delivery in a more rural or suburban setting may require traversing the street or road only once. In a rural road, mail delivery is frequently done on one side of the road, and those who live on the other side must cross the road to get their mail. In setting up the routes, though, the road could be traversed in either direction. Thus, we consider graphs with two types of connections: directed and undirected. Other postman problems [14] have been considered.