Convex and weakly convex domination in prism graphs

TL;DR

This paper investigates convex and weakly convex dominating sets in prism graphs, providing characterizations and bounds, and demonstrating that certain domination numbers can vary widely or be unbounded.

Contribution

It offers new characterizations of prism b3_{con}-fixers and -doublers and explores the unbounded differences in domination numbers between a graph and its prism.

Findings

Differences in weakly convex domination numbers can be arbitrarily large.

Convex domination number of prism graphs cannot be bounded by that of the original graph.

Characterizations of prism b3_{con}-fixers and -doublers are provided.

Abstract

For a given graph $G=(V,E)$ and permutation $\pi:V\mapsto V$ the prism $\pi G$ of $G$ is defined as follows: $V(\pi G)=V(G)\cup V(G')$, where $G'$ is a copy of $G$, and $E(\pi G)=E(G)\cup E(G')\cup M_{\pi}$, where $M_{\pi}=\{uv': u\in V(G), v=\pi(u)\}$ and $v'$ denotes the copy of $v$ in $G'$. 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 $\gamma_{con}(G).$