  The maximum stable set problem is NP-hard, even when restricted to
triangle-free graphs. In particular, one cannot expect a polynomial time
algorithm deciding if a bull-free graph has a stable set of size $k$, when $k$
is part of the instance. Our main result in this paper is to show the existence
of an FPT algorithm when we parameterize the problem by the solution size $k$.
A polynomial kernel is unlikely to exist for this problem. We show however that
our problem has a polynomial size Turing-kernel. More precisely, the hard cases
are instances of size $O(k^5)$. As a byproduct, if we forbid odd holes in
addition to the bull, we show the existence of a polynomial time algorithm for
the stable set problem. We also prove that the chromatic number of a bull-free
graph is bounded by a function of its clique number and the maximum chromatic
number of its triangle-free induced subgraphs. All our results rely on a
decomposition theorem of bull-free graphs due to Chudnovsky which is modified
here, allowing us to provide extreme decompositions, adapted to our
computational purpose.
