Recognition and Isomorphism of Proper \(\boldsymbol{H}\) -Graphs for Unicyclic \(\boldsymbol{H}\) in FPT-Time

An H-graph is an intersection graph of connected subgraphs of a suitable subdivision of a fixed graph H. Many important classes of 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 unicyclic graph, i.e. a graph which contains exactly one cycle. 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 for a fixed U, parameterized by  \(\vert U \vert \) . As an application, we obtain an FPT-time isomorphism algorithm for proper U-graphs, parameterized by  \(\vert U \vert \) . To complement this, we prove that the isomorphism problem for (proper) H-graphs is GI-complete for every fixed H which is not unicyclic nor a tree.