Worst-case efficient dominating sets in digraphs

Let $1\le n\in \mathbb{Z}$. {\it Worst-case efficient dominating sets in digraphs} are conceived so that their presence in certain strong digraphs ${\overrightarrow{ST}}_n$ corresponds to that of efficient dominating sets in star graphs $S{T}_n$: The fact that the star graphs $S{T}_n$ form a so-called dense segmental neighborly E-chain is reflected in a corresponding fact for the digraphs ${\overrightarrow{ST}}_n$. Related chains of graphs and open problems are presented as well.