A note on the independent domination number versus the domination number in bipartite graphs

TL;DR

This paper verifies Rad and Volkmann's conjecture that the ratio of the independent domination number to the domination number is at most half the maximum degree in bipartite graphs, providing extremal examples and counterexamples.

Contribution

The paper confirms the conjecture for bipartite graphs and identifies classes of graphs that attain or exceed the bound.

Findings

Conjecture holds for bipartite graphs.

Examples of extremal graphs are provided.

Graphs with odd cycles exceeding the ratio are discussed.

Abstract

Let $\gamma(G)$ and $i(G)$ be the domination number and the independent domination number of $G$, respectively. Rad and Volkmann posted a conjecture that $i(G)/ \gamma(G) \leq \Delta(G)/2$ for any graph $G$, where $\Delta(G)$ is its maximum degree (See \cite{5}: N.J. Rad, L. Volkmann, A note on the independent domination number in graphs. Discrete Appl. Math. 161(2013) 3087--3089). In this work, we verify the conjecture for bipartite graphs. Several graph classes attaining the extremal bound and graphs containing odd cycles with the ratio larger than $\Delta(G)/2$ are provided as well.