Domination and coloring are two classic problems in graph theory. The major focus of this paper is the CD-COLORING problem which combines the flavours of domination and colouring. Let $G$ be an undirected graph. A proper vertex coloring of $G$ is a $cd-coloring$ if each color class has a dominating vertex in $G$. The minimum integer $k$ for which there exists a $cd-coloring$ of $G$ using $k$ colors is called the cd-chromatic number, $\chi_{cd}(G)$. A set $S\subseteq V(G)$ is a total dominating set if any vertex in $G$ has a neighbor in $S$. The total domination number, $\gamma_t(G)$ of $G$ is the minimum integer $k$ such that $G$ has a total dominating set of size $k$. A set $S\subseteq V(G)$ is a $separated-cluster$ if no two vertices in $S$ lie at a distance 2 in $G$. The separated-cluster number, $\omega_s(G)$, of $G$ is the maximum integer $k$ such that $G$ has a separated-cluster of size $k$. In this paper, first we explore the connection between CD-COLORING and TOTAL DOMINATION. We prove that CD-COLORING and TOTAL DOMINATION are NP-Complete on triangle-free $d$-regular graphs for each fixed integer $d\geq 3$. We also study the relationship between the parameters $\chi_{cd}(G)$ and $\omega_s(G)$. Analogous to the well-known notion of `perfectness', here we introduce the notion of `cd-perfectness'. We prove a sufficient condition for a graph $G$ to be cd-perfect (i.e. $\chi_{cd}(H)= \omega_s(H)$, for any induced subgraph $H$ of $G$) which is also necessary for certain graph classes (like triangle-free graphs). Here, we propose a generalized framework via which we obtain several exciting consequences in the algorithmic complexities of special graph classes. In addition, we settle an open problem by showing that the SEPARATED-CLUSTER is polynomially solvable for interval graphs.

翻译：支配和着色是图论中的两个经典问题。本文主要研究融合了支配与着色特征的CD-着色问题。设$G$为无向图。图$G$的一个正常顶点着色称为$cd-着色$，若每种颜色类在$G$中均存在一个支配顶点。使得$G$存在使用$k$种颜色的$cd-着色$的最小整数$k$称为cd-色数，记为$\chi_{cd}(G)$。设$S\subseteq V(G)$，若$G$中任意顶点在$S$中均有邻点，则称$S$为全支配集。$G$的全支配数$\gamma_t(G)$是使得$G$存在大小为$k$的全支配集的最小整数$k$。设$S\subseteq V(G)$，若$S$中任意两个顶点在$G$中距离均不为2，则称$S$为$分离团簇$。$G$的分离团簇数$\omega_s(G)$是使得$G$存在大小为$k$的分离团簇的最大整数$k$。本文首先探讨CD-着色与全支配的联系。我们证明：对每个固定整数$d\geq 3$，CD-着色和全支配在无三角形$d$-正则图上均为NP-完全问题。同时研究参数$\chi_{cd}(G)$与$\omega_s(G)$的关系。类比经典的"完美性"概念，我们引入"cd-完美性"概念，并证明图$G$为cd-完美（即对$G$的任意诱导子图$H$有$\chi_{cd}(H)= \omega_s(H)$）的一个充分条件，该条件对某些图类（如无三角形图）也是必要的。本文提出的广义框架推导出特殊图类算法复杂性的若干重要结论。此外，我们通过证明区间图上的分离团簇问题可在多项式时间内求解，解决了一个公开问题。