  We study the following Independent Stable Set problem. Let G be an undirected
graph and M = (V(G),I) be a matroid whose elements are the vertices of G. For
an integer k\geq 1, the task is to decide whether G contains a set S\subseteq
V(G) of size at least k which is independent (stable) in G and independent in
M. This problem generalizes several well-studied algorithmic problems,
including Rainbow Independent Set, Rainbow Matching, and Bipartite Matching
with Separation. We show that
  - When the matroid M is represented by the independence oracle, then for any
computable function f, no algorithm can solve Independent Stable Set using
f(k)n^{o(k)} calls to the oracle.
  - On the other hand, when the graph G is of degeneracy d, then the problem is
solvable in time O((d+1)^kn), and hence is FPT parameterized by d+k. Moreover,
when the degeneracy d is a constant (which is not a part of the input), the
problem admits a kernel polynomial in k. More precisely, we prove that for
every integer d\geq 0, the problem admits a kernelization algorithm that in
time n^{O(d)} outputs an equivalent framework with a graph on dk^{O(d)}
vertices. A lower bound complements this when d is part of the input:
Independent Stable Set does not admit a polynomial kernel when parameterized by
k+d unless NP \subseteq coNP/poly. This lower bound holds even when M is a
partition matroid.
  - Another set of results concerns the scenario when the graph G is chordal.
In this case, our computational lower bound excludes an FPT algorithm when the
input matroid is given by its independence oracle. However, we demonstrate that
Independent Stable Set can be solved in 2^{O(k)}||M||^{O(1)} time when M is a
linear matroid given by its representation. In the same setting, Independent
Stable Set does not have a polynomial kernel when parameterized by k unless
NP\subseteq coNP/poly.
