There’s a salesman which is employed by large corporation. He have a set of $ n$ cities to visit, connected by roads, forming a graph. But as travel takes a lot of time, he has to pick hotels between visits. And he cannot take any hotel he wishes, there’s is precisely set list on $ m$ hotels.

He have to plan his travel in such way that after visiting $ p$ cities, he have to visit hotel. We may generalise it to say he have to visit $ q$ different hotels ( maybe it will be traveling celebrity problem?).

So basically he has a graph G(E,V), where E-edges, V-nodes, and two sets of nodes: C- cities, H-hotels. $ C \Union H = V$ and $ |C|=n$ , $ |H|=m$ . Find a path in a graph starting at one of the C nodes, ending at different C node and forming pattern$ (p,q)$ , $ p$ nodes from set C, then $ q$ nodes from set H, then repeat. It may not visit every H element, it may visit some of H elements many times, but it have to visit every C once.

So it is as finding Hamilton path but with “rests”.

1. Does this problem have a name? Or it is something new?
2. In what case it have solution? It probably depends both on numbers p, q, and where on the graph they are located.
3. How to find the shortest way discarding hotel costs?
4. What is an way to find optimal solution ( cheapest travel) if every hotel cost is the same.
5. What is the optimal solution when costs of hotels are different.