About K-Dependent Isolate Inclusive Sets In Graphs

Abstract

In this paper, we introduce the concept of k-dependent isolate inclusive sets in graphs and establish the necessary and sufficient conditions for a set to be a maximal k-dependent set. We then characterize 1-maximal k-dependent inclusive sets and prove that if and is a 1-maximal k-dependent inclusive set of , then forms a k-dominating set of . Furthermore, we show that if and are k-dependent vertices within , then the degree of equals the degree of . In addition, we examine the impact of vertex removal and edge removal on k-dependent isolate inclusive sets. Finally, we introduce the notion of the k-dependent isolate inclusive bondage number of a graph, defined as the minimum number of edges whose removal increases the k-dependent isolate inclusive number of the graph.