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

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 $i(G)/\gamma (G)\le \Delta (G)/2$ 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.

Accepted by Czechoslovak Mathematical Journal