Worst-case efficient dominating sets in digraphs

TL;DR

This paper explores the concept of worst-case efficient dominating sets in directed graphs, linking their properties in certain strong digraphs to those in star graphs, and discusses related graph chains and open problems.

Contribution

It introduces the notion of worst-case efficient dominating sets in digraphs and relates their existence in specific strong digraphs to properties of star graphs, extending known graph chain concepts.

Findings

Efficient dominating sets in certain strong digraphs correspond to those in star graphs.

Star graphs form a dense segmental neighborly E-chain, reflected in related digraphs.

Open problems related to graph chains are presented.

Abstract

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