Total outer-independent domination in regular graphs

Abstract

Let $G$ be a connected graph of order $n$. A set $D\subseteq V(G)$ is a total outer-independent dominating set of $G$ if $N(v)\cap D\neq \emptyset$ for every $v\in D$ and $N(v)\subseteq D$ for every $v\in V(G)\setminus D$. The total outer-independent domination number of $G$, denoted by $\gamma_{t}^{oi}(G)$, is the minimum cardinality among all total outer-independent dominating sets of $G$. We show that if $G$ is a $k$-regular graph with $k\geq 3$, then $\left(\frac{k}{2k-1}\right)n\leq \gamma_{t}^{oi}(G)\leq \left(\frac{k}{k+1}\right)n$.In addition, we characterize the $k$-regular graphs satisfying the above bounds,
except for the case of cubic graphs attaining the upper bound.
Finally, improved bounds (with respect to the previous ones) on $\gamma_{t}^{oi}(G)$
are obtained for the case in which $G$ is a claw-free regular graph.