Let $G=(V,E)$ be a graph with no isolated vertices. A vertex $v$ totally dominate a vertex $w$ ($w \ne v$), if $v$ is adjacent to $w$. A set $D \subseteq V$ called a total dominating set of $G$ if every vertex $v\in V$ is totally dominated by some vertex in $D$. The minimum cardinality of a total dominating set is the total domination number of $G$ and is denoted by $\gamma_t(G)$. A total dominator coloring of graph $G$ is a proper coloring of vertices of $G$, so that each vertex totally dominates some color class. The total dominator chromatic number $\chi_{td}(G)$ of $G$ is the least number of colors required for a total dominator coloring of $G$. The Total Dominator Coloring problem is to find a total dominator coloring of $G$ using the minimum number of colors. It is known that the decision version of this problem is NP-complete for general graphs. We show that it remains NP-complete even when restricted to bipartite, planar and split graphs. We further study the Total Dominator Coloring problem for various graph classes, including trees, cographs and chain graphs. First, we characterize the trees having $\chi_{td}(T)=\gamma_t(T)+1$, which completes the characterization of trees achieving all possible values of $\chi_{td}(T)$. Also, we show that for a cograph $G$, $\chi_{td}(G)$ can be computed in linear-time. Moreover, we show that $2 \le \chi_{td}(G) \le 4$ for a chain graph $G$ and give characterization of chain graphs for every possible value of $\chi_{td}(G)$ in linear-time.

翻译：设$G=(V,E)$为无孤立顶点的图。若顶点$v$与顶点$w$（$w \ne v$）相邻，则称$v$完全支配$w$。若集合$D \subseteq V$使得每个顶点$v\in V$都被$D$中的某个顶点完全支配，则称$D$为$G$的全支配集。全支配集的最小基数称为$G$的全支配数，记为$\gamma_t(G)$。图$G$的总支配着色是$G$顶点的一种正常着色，使得每个顶点完全支配某个颜色类。图$G$的总支配色数$\chi_{td}(G)$是实现总支配着色所需的最少颜色数。总支配着色问题旨在使用最少的颜色为$G$找到一种总支配着色。已知该问题的判定版本对一般图是NP完全的。我们证明即使限制在二分图、平面图和分裂图上，该问题仍是NP完全的。我们进一步研究了树、余图和链图等各类图的总支配着色问题。首先，我们刻画了满足$\chi_{td}(T)=\gamma_t(T)+1$的树，从而完成了对树所有可能$\chi_{td}(T)$取值的刻画。此外，我们证明了对于余图$G$，$\chi_{td}(G)$可在线性时间内计算。最后，我们证明链图$G$满足$2 \le \chi_{td}(G) \le 4$，并在线性时间内给出了$\chi_{td}(G)$取每个可能值时链图的刻画。