  We study the parameterized complexity of various classic vertex-deletion
problems such as Odd cycle transversal, Vertex planarization, and Chordal
vertex deletion under hybrid parameterizations. Existing FPT algorithms for
these problems either focus on the parameterization by solution size, detecting
solutions of size $k$ in time $f(k) \cdot n^{O(1)}$, or width
parameterizations, finding arbitrarily large optimal solutions in time $f(w)
\cdot n^{O(1)}$ for some width measure $w$ like treewidth. We unify these lines
of research by presenting FPT algorithms for parameterizations that can
simultaneously be arbitrarily much smaller than the solution size and the
treewidth.
  We consider two classes of parameterizations which are relaxations of either
treedepth of treewidth. They are related to graph decompositions in which
subgraphs that belong to a target class H (e.g., bipartite or planar) are
considered simple. First, we present a framework for computing approximately
optimal decompositions for miscellaneous classes H. Namely, if the cost of an
optimal decomposition is $k$, we show how to find a decomposition of cost
$k^{O(1)}$ in time $f(k) \cdot n^{O(1)}$. This is applicable to any graph class
H for which the corresponding vertex-deletion problem admits a constant-factor
approximation algorithm or an FPT algorithm paramaterized by the solution size.
Secondly, we exploit the constructed decompositions for solving vertex-deletion
problems by extending ideas from algorithms using iterative compression and the
finite state property. For the three mentioned vertex-deletion problems, and
all problems which can be formulated as hitting a finite set of connected
forbidden (a) minors or (b) (induced) subgraphs, we obtain FPT algorithms with
respect to both studied parameterizations.
