  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.
