A note on domination and total domination in prisms

Abstract

Recently, Azarija et al. (Electron J Combin:1.19, 2017) considered the prism $G \mathop {\square }K_2$ of a graph G and showed that $\gamma _t(G \mathop {\square }K_2) = 2\gamma (G)$ if G is bipartite, where $\gamma _t(G)$ and $\gamma (G)$ are the total domination number and the domination number of G. In this note, we give a simple proof and observe that there are similar results for other pairs of parameters. We also answer a question from that paper and show that for all graphs $\gamma _t(G \mathop {\square }K_2) \ge \frac{4}{3}\gamma (G)$, and this bound is tight.