The ratio of domination and independent domination numbers on trees

TL;DR

This paper proves a conjecture relating the ratio of independent domination to domination numbers in trees, establishing an upper bound based on the maximum degree, and identifies the extremal graph achieving this bound.

Contribution

The paper proves Rad and Volkmann's conjecture for trees and characterizes the extremal graph where the bound is tight.

Findings

The ratio i(G)/γ(G) is at most Δ(G)/2 for trees.

The extremal tree achieving the bound is characterized.

The conjecture is confirmed specifically for the class of 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.