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.

中文翻译：

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关于选择性着色问题的结构参数化

在“选择性着色”问题中，我们得到一个整数$ k $，一个图形$ G $和一个$ V（G）$划分为$ p $个部分的目标，目的是确定我们是否可以准确地选择每个部分的一个顶点，获得一个$ k $可着色的$ G $诱导子图。顶点着色的这种泛化直到最近才由Demange等人开始研究。[理论计算机科学，2014年]，出于对分布式系统上的调度问题的考虑，Guo等人。[TAMC，2020]讨论了问题的参数化复杂度的第一个结果。在这项工作中，我们研究了选择性着色的多个结构参数化。我们首先回顾Demange等人的许多硬度结果。并展示如何将它们用于为广泛使用的参数（例如路径宽度，到共聚体的距离和最大叶数）提供难处理性证明。然后，当按聚类距离进行参数化时，或在联合参数化树宽和零件数以及协树宽和零件数下进行参数化时，我们提出固定参数易处理性算法。我们的主要贡献是证明：对于每个固定的$ k \ geq 1 $，“选择性着色”在通过顶点覆盖数和零件数共同进行参数化时都不接受多项式内核，这意味着“多色独立集”不接受相同参数化下的多项式核。