  Maximum Independent Set (MIS for short) is in general graphs the paradigmatic
$W[1]$-hard problem. In stark contrast, polynomial-time algorithms are known
when the inputs are restricted to structured graph classes such as, for
instance, perfect graphs (which includes bipartite graphs, chordal graphs,
co-graphs, etc.) or claw-free graphs. In this paper, we introduce some variants
of co-graphs with parameterized noise, that is, graphs that can be made into
disjoint unions or complete sums by the removal of a certain number of vertices
and the addition/deletion of a certain number of edges per incident vertex,
both controlled by the parameter. We give a series of FPT Turing-reductions on
these classes and use them to make some progress on the parameterized
complexity of MIS in $H$-free graphs. We show that for every fixed $t \geqslant
1$, MIS is FPT in $P(1,t,t,t)$-free graphs, where $P(1,t,t,t)$ is the graph
obtained by substituting all the vertices of a four-vertex path but one end of
the path by cliques of size $t$. We also provide randomized FPT algorithms in
dart-free graphs and in cricket-free graphs. This settles the FPT/W[1]-hard
dichotomy for five-vertex graphs $H$.
