  An $H$-graph is an intersection graph of connected subgraphs of a suitable
subdivision of a fixed graph $H$. Many important classes of graphs, including
interval graphs, circular-arc graphs, and chordal graphs, can be expressed as
$H$-graphs, and, in particular, every graph is an $H$-graph for a suitable
graph $H$. An $H$-graph is called proper if it has a representation where no
subgraph properly contains another. We consider the recognition and isomorphism
problems for proper $U$-graphs where $U$ is a unicylic graph. We prove that
testing whether a graph is a (proper) $U$-graph, for some $U$, is NP-hard. On
the positive side, we give an FPT-time recognition algorithm, parametrized by
$\vert U \vert$. As an application, we obtain an FPT-time isomorphism algorithm
for proper $U$-graphs, parametrized by $\vert U \vert$. To complement this, we
prove that the isomorphism problem for (proper) $H$-graphs, is as hard as the
general isomorphism problem for every fixed $H$ which is not unicyclic.
