2-Independent Domination in Trees

Abstract

A subset $D \subseteq V(G)$ in a graph G is a dominating set if every vertex in $V(G) \setminus D$ is adjacent to at least one vertex of S. A subset $S \subseteq V(G)$ in a graph G is a 2-independent set if $\Delta (G[S]) < 2$. The 2-independence number $\alpha _{2}(G)$ is the order of a largest 2-independent set in G. Further, a subset $D \subseteq V(G)$ in a graph G is a 2-independent dominating set if D is both dominating and 2-independent. The 2-independent domination number $i^2(G)$ is the order of a smallest 2-independent dominating set in G. In this paper, we characterize all trees T of order n with $i^2(T)=\frac{n}{2}$. Moreover, we prove that for any tree T of order $n \geqslant 2$, $i^2(T) \leqslant \frac{2}{3} \alpha _{2}(T)$, and this bound is sharp.