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

Sayinath Udupa, R. S. Bhat

Research output: Contribution to journalArticle

### 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.

Original language English 2050002 Discrete Mathematics, Algorithms and Applications https://doi.org/10.1142/S1793830920500020 Accepted/In press - 01-01-2019

### Fingerprint

Independent Set
Graph in graph theory
Vertex of a graph
Cardinality
Independence number
Domination number
Dominating Set
Upper and Lower Bounds

### All Science Journal Classification (ASJC) codes

• Discrete Mathematics and Combinatorics

### Cite this

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title = "Strong vb-dominating and vb-independent sets of a graph",
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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Strong vb-dominating and vb-independent sets of a graph. / Udupa, Sayinath; Bhat, R. S.

In: Discrete Mathematics, Algorithms and Applications, 01.01.2019.

Research output: Contribution to journalArticle

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