In this paper, we study two popular variants of Graph Coloring -- Dominator Coloring and CD Coloring. In both problems, we are given a graph $G$ and a natural number $\ell$ as input and the goal is to properly color the vertices with at most $\ell$ colors with specific constraints. In Dominator Coloring, we require for each $v \in V(G)$, a color $c$ such that $v$ dominates all vertices colored $c$. In CD Coloring, we require for each color $c$, a $v \in V(G)$ which dominates all vertices colored $c$. These problems, defined due to their applications in social and genetic networks, have been studied extensively in the last 15 years. While it is known that both problems are fixed-parameter tractable (FPT) when parameterized by $(t,\ell)$ where $t$ is the treewidth of $G$, we consider strictly structural parameterizations which naturally arise out of the problems' applications. We prove that Dominator Coloring is FPT when parameterized by the size of a graph's cluster vertex deletion (CVD) set and that CD Coloring is FPT parameterized by CVD set size plus the number of remaining cliques. En route, we design a simpler and faster FPT algorithms when the problems are parameterized by the size of a graph's twin cover, a special CVD set. When the parameter is the size of a graph's clique modulator, we design a randomized single-exponential time algorithm for the problems. These algorithms use an inclusion-exclusion based polynomial sieving technique and add to the growing number of applications using this powerful algebraic technique.

翻译：本文研究图着色的两个流行变体——支配着色与CD着色。在这两个问题中，输入为一个图$G$和一个自然数$\ell$，目标是用至多$\ell$种颜色并满足特定约束条件对顶点进行正常着色。在支配着色中，要求对每个$v \in V(G)$，存在一种颜色$c$使得$v$支配所有颜色为$c$的顶点。在CD着色中，要求对每种颜色$c$，存在一个$v \in V(G)$支配所有颜色为$c$的顶点。这些问题源于社交网络和基因网络的应用，在过去十五年间得到了广泛研究。已知当参数化为$(t,\ell)$（其中$t$为$G$的树宽）时，两个问题都是固定参数可解的（FPT），但本文考虑的是问题应用中自然产生的严格结构参数化。我们证明：支配着色在参数化为图的簇顶点删除（CVD）集大小时是FPT的，而CD着色在参数化为CVD集大小加上剩余团的数量时也是FPT的。在此过程中，我们设计了一种更简单且更快的FPT算法，适用于参数化为图的孪生覆盖（一种特殊CVD集）大小的情形。当参数为图的团调子大小时，我们为这些问题设计了一种随机单指数时间算法。这些算法利用了基于容斥原理的多项式筛选技术，进一步丰富了这一强大代数技巧的应用场景。