Understanding the Independent Semitotal Domination of Graphs
0Understanding the Independent Semitotal Domination of Graphs
Layman Abstract : In graph theory, a "dominating set" is a group of selected points (vertices) in a network (graph) that help control or represent the entire structure. Researchers introduced a new concept called independent semitotal domination, which is a balance between two existing ideas: independent domination and semitotal domination.
Simply put, an independent semitotal dominating set (ISTd-set) is a group of vertices that:
Do not directly connect to each other (independent).
Still manage to influence all other vertices in the graph.
Are at least two steps away from at least one other vertex in the group.
The smallest possible size of such a set in a given graph is called the independent semitotal domination number (γit2(G)). This chapter explores when these sets exist in different types of graphs, such as wheel, helm, and barbell graphs, and calculates their corresponding independent semitotal domination numbers.
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Original Abstract : The concept of semitotal domination in a graph was introduced by Goddard, Henning, and McPillan. It enhances the idea of domination while being less strict than both total domination and weakly connected domination. In contrast, independent domination is one of the most extensively studied areas within the field of domination. A subset W⊆V(G) of a graph G is called an independent semitotal dominating set (ISTd-set) if it is an independent dominating set and each vertex in W is exactly two distances away from at least one other vertex in W. The independent semitotal domination number, γit2(G), is the smallest size of such a set. This concept lies between independent domination and semitotal domination. In this chapter, we explore independent semitotal domination in graphs, identifying the conditions under which ISTd-sets exist. Additionally, we examine ISTd-sets in some of the parameterized graphs namely wheel, helm, and barbell graphs as well as the join of graphs. As a result, we determine the corresponding independent semitotal domination numbers for these graphs.
View Book: https://doi.org/10.9734/bpi/mcsru/v3/214
#barbell_graph #helm_graph #independent_semitotal_domination #join_of_graphs #nonsingleton_independent_set #wheel_graph