  For a graph $H$, a graph $G$ is an $H$-graph if it is an intersection graph
of connected subgraphs of some subdivision of $H$. $H$-graphs naturally
generalize several important graph classes like interval or circular-arc graph.
This class was introduced in the early 1990s by B\'ir\'o, Hujter, and Tuza.
Recently, Chaplick et al. initiated the algorithmic study of $H$-graphs by
showing that a number of fundamental optimization problems are solvable in
polynomial time on $H$-graphs. We extend and complement these algorithmic
findings in several directions.
  First we show that for every fixed $H$, the class of $H$-graphs is of
logarithmically-bounded boolean-width (via mim-width). Pipelined with the
plethora of known algorithms on graphs of bounded boolean-width, this describes
a large class of problems solvable in polynomial time on $H$-graphs. We also
observe that $H$-graphs are graphs with polynomially many minimal separators.
Combined with the work of Fomin, Todinca and Villanger on algorithmic
properties of such classes of graphs, this identify another wide class of
problems solvable in polynomial time on $H$-graphs.
  The most fundamental optimization problems among the problems solvable in
polynomial time on $H$-graphs are Maximum Clique, Maximum Independent Set, and
Minimum Dominating Set. We provide a more refined complexity analysis of these
problems from the perspective of Parameterized Complexity. We show that Maximum
Independent Set and Minimum Dominating Set are W[1]-hard being parameterized by
the size of $H$ plus the size of the solution. On the other hand, we prove that
when $H$ is a tree, then Minimum Dominating Set is fixed-parameter tractable
parameterized (FPT) by the size of $H$. For Maximum Clique we show that it
admits a polynomial kernel parameterized by $H$ and the solution size.
