Find every path that passes through certain edges

I’m faced with the following problem:

Given

  • Directed and unweighted graph, where each edge E has two attributes

Goal

  • Find every path through the 3 (or more) given edges

Questions

  • Is it NP-hard?
  • Is there already an algorithm for this?
  • I was thinking about placing a node in every edge and than run Dijkstra algorithm to find the shortest path from A to B and then from B to C. Although, this complicates my graph. Any other ideas? Thanks