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