  It is known that problems like Vertex Cover, Feedback Vertex Set and Odd
Cycle Transversal are polynomial time solvable in the class of chordal graphs.
We consider these problems in a graph that has at most $k$ vertices whose
deletion results in a chordal graph, when parameterized by $k$. While this
investigation fits naturally into the recent trend of what are called
`structural parameterizations', here we assume that the deletion set is not
given.
  One method to solve them is to compute a $k$-sized or an approximate ($f(k)$
sized, for a function $f$) chordal vertex deletion set and then use the
structural properties of the graph to design an algorithm. This method leads to
at least $k^{\mathcal{O}(k)}n^{\mathcal{O}(1)}$ running time when we use the
known parameterized or approximation algorithms for finding a $k$-sized chordal
deletion set on an $n$ vertex graph.
  In this work, we design $2^{\mathcal{O}(k)}n^{\mathcal{O}(1)}$ time
algorithms for these problems. Our algorithms do not compute a chordal vertex
deletion set (or even an approximate solution). Instead, we construct a tree
decomposition of the given graph in time $2^{\mathcal{O}(k)}n^{\mathcal{O}(1)}$
where each bag is a union of four cliques and $\mathcal{O}(k)$ vertices. We
then apply standard dynamic programming algorithms over this special tree
decomposition. This special tree decomposition can be of independent interest.
  Our algorithms are adaptive (robust) in the sense that given an integer $k$,
they detect whether the graph has a chordal vertex deletion set of size at most
$k$ or output the special tree decomposition and solve the problem.
  We also show lower bounds for the problems we deal with under the Strong
Exponential Time Hypothesis (SETH).
