The dynamic traveling salesman problem, in which we assume that all objects can move with constant speed, are considered. To solve this NP-hard problem we use a game-theoretic approach and well-known solution concepts of pursuit games. In this paper, we identify the realizability areas of salesman strategies depending on the initial locations of clients. Several special cases of the problem are investigated. The paper considers several types of behavior of a traveling salesman. A Python program is made to build realizability areas of salesman strategies. Such areas are obtained for the cases of one salesman and different numbers of clients.