A 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 of $D$. The total domination number of $G$ is the minimum cardinality of any total dominating set of $G$ and is denoted by $γ_t(G)$. 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 $γ_{t,coi}(G)$. In this paper, we show that, for any tree $T$ of order $n$ and diameter at least three, $n-β(T)\leq γ_{t,coi}(T)\leq n-|L(T)|$ where $β(T)$ 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 $γ_{t,coi}(T)$ and these bounds can be arbitrarily large for some classes of trees.

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