  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 $O(\|H\|n^2)$.
