An H-graph is the intersection graph of connected subgraphs of a suitable subdivision of a fixed graph H, introduced by Biró, Hujter and Tuza (1992). We focus on S_d-graphs as a special case generalizing interval graphs. A graph G is an S_d-graph iff it is the intersection graph of connected subgraphs of a subdivision of a star S_d with d rays.

We give an FPT algorithm to solve the isomorphism problem for S_d-graphs with the parameter d. This solves an open problem of Chaplick, Töpfer, Voborník and Zeman (2016). In the course of our proof, we also show that the isomorphism problem of S_d-graphs is computationally at least as hard as the isomorphism problem of posets of bounded width.