End Super Dominating Sets in Graphs

TL;DR

This paper introduces the concept of end super dominating sets in graphs, determines their exact values for specific graph classes, and explores their applications in network server configurations.

Contribution

It defines end super dominating sets, provides exact values for certain graphs, and establishes bounds and counting methods, advancing the understanding of specialized dominating sets.

Findings

Exact end super domination numbers for specific graph classes

Tight upper bounds on end super domination number

Counting end super dominating sets in graphs

Abstract

Let $G=(V,E)$ be a simple graph. A dominating set of $G$ is a subset $S\subseteq V$ such that every vertex not in $S$ is adjacent to at least one vertex in $S$. The cardinality of a smallest dominating set of $G$, denoted by $\gamma(G)$, is the domination number of $G$. Two vertices are neighbors if they are adjacent. A super dominating set is a dominating set $S$ with the additional property that every vertex in $V \setminus S$ has a neighbor in $S$ that is adjacent to no other vertex in $V \setminus S$. Moreover if every vertex in $V \setminus S$ has degree at least~$2$, then $S$ is an end super dominating set. The end super domination number is the minimum cardinality of an end super dominating set. We give applications of end super dominating sets as main servers and temporary servers of networks. We determine the exact value of the end super domination number for specific classes of graphs, and we count the number of end super dominating sets in these graphs. Tight upper bounds on the end super domination number are established, where the graph is modified by vertex (edge) removal and contraction.