Bir{o} et al. (1992) introduced $H$-graphs, intersection graphs of connected subgraphs of a subdivision of a graph $H$. They are related to many classes of geometric intersection graphs, e.g., interval graphs, circular-arc graphs, split graphs, and chordal graphs. We negatively answer the 25-year-old question of Bir{o} et al. which asks if $H$-graphs can be recognized in polynomial time, for a fixed graph $H$. We prove that it is NP-complete if $H$ contains the diamond graph as a minor. We provide a polynomial-time algorithm recognizing $T$-graphs, for each fixed tree $T$. When $T$ is a star $S_d$ of degree $d$, we have an $O(n^{3.5})$-time algorithm. We give FPT- and XP-time algorithms solving the minimum dominating set problem on $S_d$-graphs and $H$-graphs parametrized by $d$ and the size of $H$, respectively. The algorithm for $H$-graphs adapts to an XP-time algorithm for the independent set and the independent dominating set problems on $H$-graphs. If $H$ contains the double-triangle as a minor, we prove that $H$-graphs are GI-complete and that the clique problem is APX-hard. The clique problem can be solved in polynomial time if $H$ is a cactus graph. When a graph $G$ has a Helly $H$-representation, the clique problem can be solved in polynomial time. We show that both the $k$-clique and the list $k$-coloring problems are solvable in FPT-time on $H$-graphs (parameterized by $k$ and the treewidth of $H$). In fact, these results apply to classes of graphs with treewidth bounded by a function of the clique number. We observe that $H$-graphs have at most $n^{O(|H|)}$ minimal separators which allows us to apply the meta-algorithmic framework of Fomin et al. (2015) to show that for each fixed $t$, finding a maximum induced subgraph of treewidth $t$ can be done in polynomial time. When $H$ is a cactus, we improve the bound to $O(|H|n^2)$.
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