2-Distance Zero Forcing Domination: Properties and Realization Problems.

Source abstract

Let G be a graph. Then A \subseteq V(G) is called a 2-distance zero forcing dominating if A is both a 2-distance zero forcing and dominating set of G. The minimum cardinality among all 2-distance zero forcing dominating sets of G, denoted by \gamma _{Z^2} (G), is the 2-distance zero forcing domination number of G. In this study, we give some properties of this new parameter and investigate its connections with 2-distance zero forcing and domination, respectively. We show that 2-distance zero forcing domination is at least equal to the 2-distance zero forcing (resp. domination) number of any graph. Moreover, we give a realization result involving a 2-distance zero forcing domination and a domination concepts, and subsequently show that the difference between these two parameters can be made arbitrarily large. Moreover, we characterize 2-distance zero forcing dominating sets in some special graphs and the join of any two graphs. Finally, we derive the exact values or bounds of the parameter.