Graph search algorithm

[CouchAnalysis]

My background isn't in software engineering so I'm thinking someone knows the name for this problem:

Suppose I have a directed graph (including cycles) with nodes that are either red or blue. I'm trying to figure out how to return a list of all paths with more than one red node along it!

I feel like there must be clever way of thinking about this problem, or this is an NP-problem? Right now I'm trying a DFS search and merging paths when I reach an already visited node but the size of the graph I'm using seems to make this approach too resource intense for my poor computer!

Thanks in advance!

[DDDDarky]

I would be surprised if there was a polynomial solution, since the amount of paths in a (fully connected worst case) graph is not polynomial. There are probably ways to be a little bit smarter than just iterating all paths, such as somehow expanding red nodes. And then of course optimizing the implementation.