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.

一组\（S \）的曲线图的（顶点）\（G \）被称为控制集的\（G \）如果在每个顶点\（V - S \）相邻于一个节点\（S \）。控制集\（S \） ，例如通过诱导的子图\（S \）已分离的顶点被称为分离物支配集，并且还分离物支配集的最小计数被称为的分离控制数\（G \ ），并用\（\ gamma_ {is}（G）\）表示。如果\（X-E \）中的每个边都与\（S \）中的某个边相邻，则子集\（X \ subseteq E \）被称为边控制集。。边控制数是\（G \）的最小边控制集的计数，并由\（\ gamma ^ {\ prime} \）使用。如果不在\（X \）中的每个边缘都恰好与\（X \）中的一个边缘相邻，则\（E \）的边缘\（X \）的集合被称为是理想的边缘控制集。理想的边缘支配数是最小基数采用\（G \）的完美边缘支配集，并用\（\ gamma_ {p} ^ {^ {\ prime}} \）表示。在本文中，我们启动了与孤立统治相关的束缚的调查。隔离束缚数\（b_ {is}（G）\）概述为满足\（\ gamma_ {is}（G ^ {\ prime}的图\（G ^ {\ prime} \）中结束于\（G \）的边集合的最小基数）> \ gamma_ {是}（G）\）。我们获得了一些关于分离物支配集的结果，以及分离物结合数的相同值。此外，我们研究了分离株键合数的一些界限，这个界限很敏锐，并分析了在什么条件下支配参数和分离株支配参数相等。此外，我们还发现了一些其他的结果，可以完美地控制边缘，并且对\（\ gamma ^ {\ prime} = \ gamma_ {p} ^ {^ {\ prime}} \）的树进行了表征，并得出了更令人兴奋的结果。