  An $H$-graph is one representable as the intersection graph of connected
subgraphs of a suitable subdivision of a fixed graph $H$, introduced by
Bir\'{o}, Hujter and Tuza (1992). An $H$-graph is proper if the representing
subgraphs of $H$ can be chosen incomparable by the inclusion. In this paper, we
focus on the isomorphism problem for $S_d$-graphs and $T$-graphs, where $S_d$
is the star with $d$ rays and $T$ is an arbitrary fixed tree.
  Answering an open problem of Chaplick, T\"{o}pfer, Voborn\'{\i}k and Zeman
(2016), we provide an FPT-time algorithm for testing isomorphism and computing
the automorphism group of $S_d$-graphs when parameterized by~$d$, which
involves the classical group-computing machinery by Furst, Hopcroft, and Luks
(1980). We also show that the isomorphism problem of $S_d$-graphs is at least
as hard as the isomorphism problem of posets of bounded width, for which no
efficient combinatorial-only algorithm is known to date. Then we extend our
approach to an XP-time algorithm for isomorphism of $T$-graphs when
parameterized by the size of $T$. Lastly, we contribute a simple FPT-time
combinatorial algorithm for isomorphism testing in the special case of proper
$S_d$- and $T$-graphs.
