  We study the tractability of the maximum independent set problem from the
viewpoint of graph width parameters, with the goal of defining a width
parameter that is as general as possible and allows to solve independent set in
polynomial-time on graphs where the parameter is bounded. We introduce two new
graph width parameters: one-sided maximum induced matching-width (o-mim-width)
and neighbor-depth. O-mim-width is a graph parameter that is more general than
the known parameters mim-width and tree-independence number, and we show that
independent set and feedback vertex set can be solved in polynomial-time given
a decomposition with bounded o-mim-width. O-mim-width is the first width
parameter that gives a common generalization of chordal graphs and graphs of
bounded clique-width in terms of tractability of these problems.
  The parameter o-mim-width, as well as the related parameters mim-width and
sim-width, have the limitation that no algorithms are known to compute
bounded-width decompositions in polynomial-time. To partially resolve this
limitation, we introduce the parameter neighbor-depth. We show that given a
graph of neighbor-depth $k$, independent set can be solved in time $n^{O(k)}$
even without knowing a corresponding decomposition. We also show that
neighbor-depth is bounded by a polylogarithmic function on the number of
vertices on large classes of graphs, including graphs of bounded o-mim-width,
and more generally graphs of bounded sim-width, giving a quasipolynomial-time
algorithm for independent set on these graph classes. This resolves an open
problem asked by Kang, Kwon, Str{\o}mme, and Telle [TCS 2017].
