In this paper, we investigate the \textsc{Grundy Coloring} problem for graphs with a cluster modulator, a structure commonly found in dense graphs. The Grundy chromatic number, representing the maximum number of colors needed for the first-fit coloring of a graph in the worst-case vertex ordering, is known to be $W[1]$-hard when parameterized by the number of colors required by the most adversarial ordering. We focus on fixed-parameter tractable (FPT) algorithms for solving this problem on graph classes characterized by dense substructures, specifically those with a cluster modulator. A cluster modulator is a vertex subset whose removal results in a cluster graph (a disjoint union of cliques). We present FPT algorithms for graphs where the cluster graph consists of one, two, or $k$ cliques, leveraging the cluster modulator's properties to achieve the best-known FPT runtimes, parameterized by both the modulator's size and the number of cliques.

翻译：本文研究了具有聚类调节器（稠密图中常见结构）的图的\textsc{Grundy着色}问题。Grundy色数表示在最坏顶点排序下对图进行首次适应着色所需的最大颜色数，已知在由最坏排序所需颜色数参数化时是$W[1]$-困难的。我们专注于在具有稠密子结构特征的图类上解决此问题的固定参数可处理（FPT）算法，特别是那些具有聚类调节器的图类。聚类调节器是指移除后得到聚类图（团的不交并）的顶点子集。我们针对聚类图包含一个、两个或$k$个团的情况提出了FPT算法，利用聚类调节器的特性实现了目前已知最佳的FPT运行时间，其参数化同时基于调节器规模和团的数量。