Characterizations of $J$-Total Dominating Sets in Some Special Graphs and Graphs under Some Operations

Abstract

Let G be a graph with no isolated vertex. A subset M ⊆ V (G) is called a J-open set if NG(a)\NG(b) ̸= ∅ and NG(b)\NG(a) ̸= ∅ ∀ a, b ∈ M, where a ̸= b. If in addition, M is a total dominating in G, then we call M a J-total dominating set in G. The maximum cardinality among
all J-total dominating set in G, denoted by γJt(G), is called the J-total domination number of G. In this paper, we characterize J-total dominating sets in some special graphs and join of two graphs, and we use these results to obtain formulas for the parameters of these graphs. Moreover, we determine its relationships with other known parameters in graph theory. Finally, we derive the lower bound of the parameter for the corona of two graphs.