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Let $G=(V,E)$ be a graph with no isolated vertices. A vertex $v$ totallydominate 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 $v\in V$ istotally dominated by some vertex in $D$. The minimum cardinality of a totaldominating 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 ofvertices of $G$, so that each vertex totally dominates some color class. Thetotal dominator chromatic number $\chi_{td}(G)$ of $G$ is the least number ofcolors required for a total dominator coloring of $G$. The Total DominatorColoring problem is to find a total dominator coloring of $G$ using the minimumnumber of colors. It is known that the decision version of this problem isNP-complete for general graphs. We show that it remains NP-complete even whenrestricted to bipartite, planar and split graphs. We further study the TotalDominator Coloring problem for various graph classes, including trees, cographsand chain graphs. First, we characterize the trees having$\chi_{td}(T)=\gamma_t(T)+1$, which completes the characterization of treesachieving all possible values of $\chi_{td}(T)$. Also, we show that for acograph $G$, $\chi_{td}(G)$ can be computed in linear-time. Moreover, we showthat $2 \le \chi_{td}(G) \le 4$ for a chain graph $G$ and give characterizationof chain graphs for every possible value of $\chi_{td}(G)$ in linear-time.

Complexity of total dominator coloring in graphs