note on total co-independent domination in trees

A set $D$ of vertices of a graph $G$ is a total dominating set if every vertex of $G$ is adjacent to at least one vertex in $D$. The total dominating set $D$ is called a total co-independent dominating set if $V(G)\setminus D$ is an independent set and has at least one vertex. The minimum cardinality of any total co-independent dominating set is denoted by $\gamma_{t,coi}(G)$. In this paper, we show that, for any tree $T$ of order $n$ and diameter at least three, $n-\beta(T)\leq \gamma_{t,coi}(T)\leq n-|L(T)|$ where $\beta(T)$ is the maximum cardinality of any independent set and $L(T)$ is the set of leaves of $T$. We also characterize the families of trees attaining the extremal bounds above and show that the differences between the value of $\gamma_{t,coi}(T)$ and these bounds can be arbitrarily large for some classes of trees.