Total complementary acyclic domination in graphs

Let $G=(V,E)$ be a graph without isolates. A subset $S$ of $V(G)$ is called a total dominating set of $G$ if for every $vin V$, there exists $uin S$ such that $u$ and $v$ are adjacent. $S$ is called a total complementary acyclic dominating set of $G$, if $S$ is a total dominating set of $G$ and $leftlangle V-S ight angle$ is acyclic.$V(G)$ is a total complementary acyclic dominating set of $G$ (since $G$ has no isolates). The minimum cardinality of a total complementary acyclic dominating set of $G$ is called the total complementary acyclic domination number of $G$ and is denoted by $gamma_{c-a}^{t}(G)$. In this paper, characterization of graphs for which $gamma_{c-a}^{t}(G)$ takes specific values are found.