  In this paper, we investigate the complexity of Maximum Independent Set (MIS)
in the class of $H$-free graphs, that is, graphs excluding a fixed graph as an
induced subgraph. Given that the problem remains $NP$-hard for most graphs $H$,
we study its fixed-parameter tractability and make progress towards a dichotomy
between $FPT$ and $W[1]$-hard cases. We first show that MIS remains $W[1]$-hard
in graphs forbidding simultaneously $K_{1, 4}$, any finite set of cycles of
length at least $4$, and any finite set of trees with at least two branching
vertices. In particular, this answers an open question of Dabrowski et al.
concerning $C_4$-free graphs. Then we extend the polynomial algorithm of
Alekseev when $H$ is a disjoint union of edges to an $FPT$ algorithm when $H$
is a disjoint union of cliques. We also provide a framework for solving several
other cases, which is a generalization of the concept of \emph{iterative
expansion} accompanied by the extraction of a particular structure using
Ramsey's theorem. Iterative expansion is a maximization version of the
so-called \emph{iterative compression}. We believe that our framework can be of
independent interest for solving other similar graph problems. Finally, we
present positive and negative results on the existence of polynomial (Turing)
kernels for several graphs $H$.
