A subset $S\subseteq V$ in a graph $G=(V,E)$ is a $k$-quasiperfect dominating set (for $k\geq1$) if every vertex not in $S$ is adjacent to at least one and at most $k$ vertices in $S$. The cardinality of a minimum $k$-quasiperfect dominating set in $G$ is denoted by $\gamma_{1k}(G)$. Those sets were first introduced by Chellali et al. (2013) as a generalization of the perfect domination concept and allow us to construct a decreasing chain of quasiperfect dominating numbers $n\geq\gamma_{11}(G)\geq\gamma_{12}(G)\geq\dots\geq\gamma_{1\Delta}(G)=\gamma(G)$ in order to indicate how far is $G$ from being perfectly dominated. In this paper we study properties, existence and realization of graphs for which the chain is short, that is, $\gamma_{12}(G)=\gamma(G)$. Among them, one can find cographs, claw-free graphs and graphs with extremal values of $\Delta(G)$.