  We introduce a one-sided incidence tree decomposition of a CNF $\varphi$.
This is a tree decomposition of the incidence graph of $\varphi$ where the
underlying tree is rooted and the set of bags containing each clause induces a
directed path in the tree. The one-sided treewidth is the smallest width of a
one-sided incidence tree decomposition.
  We consider a class of unsatisfiable CNF $\varphi$ that can be turned into
one of one sided treewidth at most $k$ by removal of at most $p$ clauses. We
show that the size of regular resolution for this class of CNFs is FPT
parameterized by $k$ and $p$. The results contributes to understanding the
complexity of resolution for CNFs of bounded incidence treewidth, an open
problem well known in the areas of proof complexity and knowledge compilation.
In particular, the result significantly generalizes all the restricted classes
of CNFs of bounded incidence treewidth that are known to admit an FPT sized
resolution.
  The proof includes an auxiliary result and several new notions that may be of
an independent interest.
