J-Domination in Graphs

Let G be a graph. A subset D = {d1, d2, · · · , dm} of vertices of G is called a J-set ifNG[di] \ NG[dj ] ̸= ∅ for every i ̸= j, where i, j ∈ {1, 2, . . . , m}. A J-set is called a J-dominatingset of G if D = {d1, d2, . . . , dm} is a dominating set of G. The J-domination number of G, denotedby γJ (G), is the maximum cardinality of a J-dominating set of G. In this paper, we introducethis new concept and we establish formulas and properties on some classes of graphs and in joinof two graphs. Upper and lower bounds of J-domination parameter with respect to the order of agraph and other parameters in graph theory are obtained. In addition, we present realization resultinvolving this parameter and the standard domination. Moreover, we characterize J-dominatingsets in some classes of graphs and join of two graphs and finally determine the exact value of theparameter of each of these graphs.