Strong vb-dominating and vb-independent sets of a graph

Let G = (V,E) be a graph. A vertex u V strongly (weakly) b-dominates block b B(G) if dvb(u) ≥ dvb(w) (dvb(u) ≤ dvb(w)) for every vertex w in the block b. A set S V is said to be strong (weak) vb-dominating set (SVBD-set) (WVBD-set) if every block in G is strongly (weakly) b-dominated by some vertex in S. The strong (weak) vb-domination number γsvb = γsvb(G) (γwvb = γwvb(G)) is the order of a minimum SVBD (WVBD) set of G. A set S > V is said to be strong (weak) vertex block independent set (SVBI-set (WVBI-set)) if S is a vertex block independent set and for every vertex u S and every block b incident on u, there exists a vertex w V-S in the block b such that dvb(u) ≥ dvb(w) (dvb(u) ≤ dvb(w)). The strong (weak) vb-independence number βsvb = βsvb(G) (βwvb = βwvb(G)) is the cardinality of a maximum strong (weak) vertex block independent set (SVBI-set) (WVBI-set) of G. In this paper, we investigate some relationships between these four parameters. Several upper and lower bounds are established. In addition, we characterize the graphs attaining some of the bounds.