We study the computational complexity of the H-free Star System problem. We prove that when H is a path or a cycle on at most four vertices the problem is polynomial time solvable.
In complement to this result, we show that if H belongs to a certain large class of graphs the H-free Star System problem is NP-complete. In particular, the problem is NP-complete when H is either a cycle or a path on at least five vertices.
This yields a complete dichotomy for paths and cycles.