Total complementary tree domination in grid graphs

Let $G = (V, E)$ be a nontrivial, simple, finite and undirected graph. A dominating set $D$ is called a complementary tree dominating set if the induced subgraph $$ is a tree. The minimum cardinality of a complementary tree dominating set is called the complementary tree domination number of $G$ and is denoted by $gamma_{ctd}(G)$. A dominating set $D_t$ is called a total complementary tree dominating set if every vertex $v in V$ is adjacent to an element of $D_t$ and $$ is a tree. The minimum cardinality of a total complementary tree dominating set is called the total complementary tree domination number of $G$ and is denoted by $gamma_{tctd}$. In this paper, we determine the total complementary tree domination numbers of some grid graph.