Double outer-independent domination number of graphs

Abstract

Let G be a graph with no isolated vertex. A set D ⊆ V(G) is a double outer-independent dominating set of G if V (G) \ D is an independent set and | N[υ] ∩ D| ≥ 2 for every v ϵ V(G), where N[υ] denotes the closed neighbourhood of v. The minimum cardinality among all double outer-independent dominating sets of G is the double outer-independent domination number of G. In this paper, we continue with the study of this parameter. For instance, we give some relationships that exist between this parameter and other domination invariants in graph. Also, we present tight bounds and show some classes of graphs for which the bounds are achieved. Finally, we provide closed formulas for the double outer-independent domination number of rooted product graphs, and characterize the graphs reaching these expressions.