### 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 |
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Article number | 2050002 |

Journal | Discrete Mathematics, Algorithms and Applications |

DOIs | |

Publication status | Accepted/In press - 01-01-2019 |

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### All Science Journal Classification (ASJC) codes

- Discrete Mathematics and Combinatorics

### Cite this

*Discrete Mathematics, Algorithms and Applications*, [2050002]. https://doi.org/10.1142/S1793830920500020