I'm looking for some search terms/papers that I can read for a problem like the following.
I have some undirected graph, G, where there are non-negative weights between nodes. However, each node has an associated "state"/"colour". Changing any node's state/colour changes the weights between adjacent nodes.
I need find the set of colours that minimises the total weight of the tree, but I'm having trouble matching this to an existing known problem. I've considered duplicating the nodes,once for each state - but its not quite a minimum spanning tree problem, as not all nodes then need to (or indeed can) be visited.
Any ideas as to existing algorithms or papers for this problem? I'm having trouble coming up with good search terms.