What algorithm can I use to find the shortest path between specified node types in a graph?

Question

This is the problem:

I have n points (p1, p2, p3, .. pn), each of them can connect to any other with a determined cost x.

Each point belongs to one of a set of p

Answer 1

On the revision of your question it seems you ask for one node per letter - in that case it is a simple dynamic programming solution: Calculate all the shortest paths of length 1, which satisfy the beginning of your sequence, between each pair of nodes. Then having for k all such paths for all node pairs, it is trivial to construct for k+1.