Understanding the Independent Semitotal Domination of Graphs

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.