On Unique Realizations of Domination Chain Parameters

Abstract

The domination chain ${\iota }_r(G)\le \gamma (G)\le \iota (G)\le {\beta }_o(G)\le \mathrm{\Gamma }(G)\le IR(G)$, which holds for any graph $G$, is the subject of much research. In this paper, we consider the maximum number of edges in a graph having one of these domination chain parameters equal to $2$ through a unique realization. We show that a specialization of the domination chain still holds in this setting.