A new lower bound for the independent domination number of a tree

Abstract

A set $D$ of vertices in a graph $G$ is an independent dominating set of $G$ if $D$ is an independent set and every vertex not in $D$ is adjacent to a vertex in $D$. The independent domination number of $G$, denoted by $i(G)$, is the minimum cardinality among all independent dominating sets of $G$. In this paper we show that if $T$ is a nontrivial tree, then $i(T)\geq \frac{n(T)+\gamma(T)-l(T)+2}{4}$, where $n(T)$, $\gamma(T)$ and $l(T)$ represent the order, the domination number and the number of leaves of $T$, respectively. In addition, we characterize the trees achieving this new lower bound.