Udupa, Sayinath and Bhat, R S (2020) Strong vb-dominating and vb-independent sets of a graph. Discrete Mathematics, algorithms and applications, 12 (1). ISSN 1793-8309

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Abstract

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.

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