The ratio of domination and independent domination numbers on trees

Abstract

Let $\gamma(G)$ and $i(G)$ be the domination number and the independent domination number of $G$, respectively. In 1977, Hedetniemi and Mitchell began with the comparison of of $i(G)$ and $\gamma(G)$ and recently Rad and Volkmann posted a conjecture that $i(G)/ \gamma(G) \leq \Delta(G)/2$, where $\Delta(G)$ is the maximum degree of $G$. In this work, we prove the conjecture for trees and provide the graph achieved the sharp bound.