The Outer independent $\{2\}$-domination in trees

Outer independent $\{2\}$-domination in trees

Abstract

An outer independent $\{2\}$-dominating function (OI$\{2\}$D-function) of a
graph $G$ is a function $f:V(G)\longrightarrow\{0,1,2\}$ such that no two
vertices assigned 0 under $f$ are adjacent, and $f(N[v])\geq2$ for all $v\in
V(G),$ where $N[v]$ stands for the set of neighbors of $v$ plus $v.$ The
weight of an OI$\{2\}$D-function is the value $\omega(f)=\Sigma_{u\in
V(G)}f(u)$, and the minimum weight of an OI$\{2\}$D-function of $G$ is the
outer independent $\{2\}$-domination number $\gamma_{oi\{2\}}(G)$ of $G$. In
this paper, we first determine the exact value of the outer
independent-$\{2\}$-domination number for perfect binary trees, an then we
provide a lower bound and an upper bound for the outer independent
$\{2\}$-domination number for trees in terms of the covering number, the
independence number, the number of leaves and the number of stems (support
vertices)