Final answer:
The cities where a traveler can start to traverse each road once follow the rules of an Eulerian path or circuit in graph theory. An Eulerian path requires zero or two vertices with an odd number of edges (odd degree), while an Eulerian circuit requires all vertices to have an even degree.
Step-by-step explanation:
The student has described a network problem that falls under graph theory, a part of mathematics that studies graphs, which in this context refers to a collection of points (vertices) connected by lines (edges). The task described resembles finding a path that traverses each edge exactly once, which is known as an Eulerian path.
An Eulerian path exists if and only if the graph is connected and has exactly zero or two vertices with an odd degree (an odd number of edges). If there are no vertices with an odd degree, then an Eulerian circuit exists, and the trail can start and end at the same vertex. If there are two vertices with an odd degree, then the Eulerian path must start at one of these vertices and end at the other.
In this specific case, the student is looking for cities where the traveler can start such that every road segment is traveled once. Without the actual graph or its degree sequence, we cannot name the specific cities. However, based on the information given regarding Eulerian paths, the cities that qualify are those represented by vertices with an odd degree (if any), or any city if the graph supports an Eulerian circuit.