On Cartesian Products Having a Minimum Dominating Set that is a Box or a Stairway

Abstract

In this note we characterize the pairs of graphs $G$ and $H$, for which $\gamma (G\,\Box \,H)$ equals $\min \{|V(G)|,|V(H)|\}$. Notably, assuming that $|V(G)|\le |V(H)|$, $G$ can be an arbitrary graph, and $H$ is a join $L\oplus F$, where $L$ is any spanning supergraph of the graph $\mathcal{L}(G:A_1,\ldots , A_{\ell })$, which is determined by a partition $(A_1,\ldots , A_{\ell })$ of $V(G)$ and $F$ is any graph such that $|V(F)|\ge |V(G)|-\ell$. Furthermore we give some sufficient and some necessary conditions for pairs of graphs $G$ and $H$ to satisfy $\gamma (G\,\Box \,H)=\min \{\gamma (G)|V(H)|, |V(G)|\gamma (H)\}.$