  Bir\'{o}, Hujter, and Tuza introduced the concept of $H$-graphs (1992),
intersection graphs of connected subgraphs of a subdivision of a graph $H$.
They naturally generalize many important classes of graphs, e.g., interval
graphs and circular-arc graphs. We continue the study of these graph classes by
considering coloring, clique, and isomorphism problems on $H$-graphs.
  We show that for any fixed $H$ containing a certain 3-node, 6-edge multigraph
as a minor that the clique problem is APX-hard on $H$-graphs and the
isomorphism problem is isomorphism-complete. We also provide positive results
on $H$-graphs. Namely, when $H$ is a cactus the clique problem can be solved in
polynomial time. Also, when a graph $G$ has a Helly $H$-representation, the
clique problem can be solved in polynomial time. Finally, we observe that one
can use treewidth techniques to show that both the $k$-clique and list
$k$-coloring problems are FPT on $H$-graphs. These FPT results apply more
generally to treewidth-bounded graph classes where treewidth is bounded by a
function of the clique number.
