Convex and weakly convex domination in prism graphs

Abstract

For a given graph $G=(V,E)$ and permutation $\pi :V\mapsto V$ the prism $\pi G$ of $G$ is defined as follows: $V(\pi G)=V(G)\cup V(G^{\prime })$, where $G^{\prime }$ is a copy of $G$, and $E(\pi G)=E(G)\cup E(G^{\prime })\cup M_{\pi }$, where $M_{\pi }=\{u{v}^{\prime }:u\in V(G),v=\pi (u)\}$ and $v^{\prime }$ denotes the copy of $v$ in $G^{\prime }$. We study and compare the properties of convex and weakly convex dominating sets in prism graphs. In particular, we characterize prism ${\gamma }_{con}$-fixers and -doublers. We also show that the differences ${\gamma }_{wcon}(G)-{\gamma }_{wcon}(\pi G)$ and ${\gamma }_{wcon}(\pi G)-2{\gamma }_{wcon}(G)$ can be arbitrarily large, and that the convex domination number of $\pi G$ cannot be bounded in terms of ${\gamma }_{con}(G).$