Total complementary tree domination in grid graphs

TLDR: In this article, the authors determined the total complementary tree domination number of a tree dominating set in a grid graph, i.e., the minimum cardinality of a dominating set for which every vertex is adjacent to an element of the dominating set and every subgraph is a tree.

Abstract: 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.