On the number of isolate dominating sets of certain graphs

TL;DR

This paper investigates the count of isolate dominating sets in specific graphs, focusing on the properties and enumeration of such sets related to the graph's domination number.

Contribution

It introduces the concept of isolate dominating sets and provides enumeration results for these sets in certain classes of graphs.

Findings

Derived formulas for the number of isolate dominating sets in specific graphs

Established bounds for the isolate domination number

Analyzed the relationship between domination number and isolate domination number

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 $G$. A dominating set $S$ is an isolate dominating set of $G$, if the induced subgraph $G[S]$ has at least one isolated vertex. The isolate domination number, $\gamma_0(G)$, is the minimum cardinality of an isolate dominating set of $G$. In this paper, we count the number of isolate dominating sets of some specific graphs.