Let \(G=(V,E)\) be a graph with no isolated vertices. A vertex v totally dominates 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 \(\chi _{{\textrm{td}}}(G)\) 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 \(\chi _{{\textrm{td}}}(T)=\gamma _t(T)+1\), which completes the characterization of trees achieving all possible values of \(\chi _{{\textrm{td}}}(T)\). Also, we show that for a cograph G, \(\chi _{{\textrm{td}}}(G)\) can be computed in linear-time. Moreover, we show that \(2 \le \chi _{{\textrm{td}}}(G) \le 4\) for a chain graph G and then we characterize the class of chain graphs for every possible value of \(\chi _{{\textrm{td}}}(G)\) in linear-time.

令G=(V,E)\)为没有孤立顶点的图。如果v与w相邻，则顶点v完全支配顶点w ( \(w \ne v\) )。如果每个顶点\ (v\in V\)完全被D中的某个顶点支配，则集合 \( D \subseteq V \)称为G的总支配集。总支配集的最小基数是G的总支配数，用\(\gamma_t(G)\)表示。图G的总支配着色是G的顶点的适当着色，以便每个顶点完全支配某个颜色类。G 的总主色数\(\chi _{{\textrm{ td }}}(G)\)是G的总主色着色所需的最少颜色数。总支配着色问题是使用最少颜色数找到G的总支配着色。众所周知，该问题的决策版本对于一般图来说是 NP 完全的。我们证明，即使仅限于二分图、平面图和分裂图，它仍然是 NP 完全的。我们进一步研究各种图类的总支配着色问题，包括树、图和链图。首先，我们表征具有\(\chi _{{\textrm{td}}}(T)=\gamma _t(T)+1\) 的树，这完成了对实现\(\ chi _{{\textrm{td}}}(T)\)。此外，我们还表明，对于 cograph G，\(\chi _{{\textrm{td}}}(G)\)可以在线性时间内计算。此外，我们证明链图G的 \(2 \le \chi _{{\textrm{td}}}(G) \le 4\) ，然后我们针对每个可能的值来表征链图的类(\chi _{{\textrm{td}}}(G)\)线性时间。