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An $H$-graph is an intersection graph of connected subgraphs of a suitablesubdivision of a fixed graph $H$. Many important classes of graphs, includinginterval graphs, circular-arc graphs, and chordal graphs, can be expressed as$H$-graphs, and, in particular, every graph is an $H$-graph for a suitablegraph $H$. An $H$-graph is called proper if it has a representation where nosubgraph properly contains another. We consider the recognition and isomorphismproblems for proper $U$-graphs where $U$ is a unicylic graph. We prove thattesting whether a graph is a (proper) $U$-graph, for some $U$, is NP-hard. Onthe positive side, we give an FPT-time recognition algorithm, parametrized by$\vert U \vert$. As an application, we obtain an FPT-time isomorphism algorithmfor proper $U$-graphs, parametrized by $\vert U \vert$. To complement this, weprove that the isomorphism problem for (proper) $H$-graphs, is as hard as thegeneral isomorphism problem for every fixed $H$ which is not unicyclic.

Recognition and Isomorphism of Proper $\boldsymbol{U}$-graphs in FPT-time