Adjoining edges to $G\square H$ to construct a minimal dominating set of size $\gamma (G)\gamma (H)$

Abstract

For graphs $G,H$ it is possible to add $(|V(G)|-\gamma (G))(|V(H)|-\gamma (H))$ edges to the Cartesian product $G\square H$ such that a minimal dominating set $D$ of size $\gamma (G)\gamma (H)$ emerges. We hypothesize that $D$ is also a minimum dominating set for the resulting graph and show that this implies Vizing's conjecture.