  In the Selective Coloring problem, we are given an integer $k$, a graph $G$,
and a partition of $V(G)$ into $p$ parts, and the goal is to decide whether or
not we can pick exactly one vertex of each part and obtain a $k$-colorable
induced subgraph of $G$. This generalization of Vertex Coloring has only
recently begun to be studied by Demange et al. [Theoretical Computer Science,
2014], motivated by scheduling problems on distributed systems, with Guo et al.
[TAMC, 2020] discussing the first results on the parameterized complexity of
the problem. In this work, we study multiple structural parameterizations for
Selective Coloring. We begin by revisiting the many hardness results of Demange
et al. and show how they may be used to provide intractability proofs for
widely used parameters such as pathwidth, distance to co-cluster, and max leaf
number. Afterwards, we present fixed-parameter tractability algorithms when
parameterizing by distance to cluster, or under the joint parameterizations
treewidth and number of parts, and co-treewidth and number of parts. Our main
contribution is a proof that, for every fixed $k \geq 1$, Selective Coloring
does not admit a polynomial kernel when jointly parameterized by the vertex
cover number and the number of parts, which implies that Multicolored
Independent Set does not admit a polynomial kernel under the same
parameterization.
