How can I compute a route/path of a certain length between two places?

Given two places in a specific region, how can I compute a path that is of certain distance (the distance is given as a parameter) which connects the two places. The resulting path should ideally be composed of roads that are visited no more than once.

I have learnt that this problem can be seen as a graph problem where the roads in the map represent the edges of the graph and the intersection/junctions represent the nodes of the graph. From this we can simplify the problem to finding a path of certain length for any two nodes in the graph.

One of my approaches to this problem can be broken into two steps:

• Finding the shortest path between the two nodes (using A* or a similar algorithm).
• Extending/Expanding the shortest path that is returned from A* so that it's long enough.

Now, I am not exactly sure if this approach is worth pursuing or if a better approach exists? Also, I've not come across any methods that would allow me to expand the path - are there any specific techniques that can achieve this?