RETRACTED ARTICLE: An investigation of unicyclic graphs in which the isolate bondage number is equal to three in graph network theory

Abstract

A set $S$ (of vertices) of a graph $G$ is termed a dominating set of $G$ if each vertex in $V - S$ is adjacent to a node in $S$. A dominating set $S$ such as the subgraph induced by $S$ has an isolated vertex is termed an isolate dominating set and also the minimum count of an isolate dominating set is termed the isolate domination number of $G$ and it is represented by $\gamma_{is} (G)$. A subset $X \subseteq E$ is said to be an edge dominating set if each edge in $X - E$ is adjacent to some edge in $S$. The edge domination number is that the count of the smallest edge dominating set of $G$ and is employed by $\gamma^{\prime}$. A collection of edges $X$ of $E$ is claimed to be a perfect edge dominating set if each edge not in $X$ is adjacent to precisely one edge in $X$. The ideal edge domination number is that the minimum cardinality has taken perfect edge dominating sets of $G$ and is denoted by $\gamma_{p}^{^{\prime}}$. In this paper, we initiate the survey of bondage related to isolate domination. The isolate bondage number $b_{is} (G)$ is outlined to be the minimum cardinality of a collection of edges whose relieved from $G$ ends up in a graph $G^{\prime}$ fulfilling $\gamma_{is} (G^{\prime}) > \gamma_{is} (G)$. We obtain several results for isolate dominating set and identical values of isolate bondage number. Moreover, we investigate some bounds for the isolate bondage number, and this bound is keen and analyze under which conditions the domination parameter and isolate domination parameter are equal. Also, we found some more results for perfect edge domination, and we characterize trees for which $\gamma^{\prime} = \gamma_{p}^{^{\prime}}$ and further exciting results.