### Abstract

A vertex v in a graph G = (V, E) is strong (weak) if deg (v) ≥ deg (u)(deg (v) ≤ deg (u)) for every u adjacent to v in G. A set S ⊆ V is said to be strong (weak) if every vertex in S is a strong (weak) vertex in G. A strong (weak) set which is independent is called a strong independent set [SIS] (weak independent set [WIS]). The strong (weak) independence numbers α = s α (G) (w α = w α (G)) is the maximum cardinality of an SIS (WIS). For an edge x = uv, v strongly covers the edge x if deg (v) ≥ deg (u) in G. Then u weakly covers x. A set S ⊆ V is a strong vertex cover [SVC] (weak vertex cover [WVC]) if every edge in G is strongly (weakly) covered by some vertex in S. The strong (weak) vertex covering numbers β = s β (G)(w β = w β (G)) is the minimum cardinality of an SVC (WVC). In this paper, we investigate some relationships among these four new parameters. For any graph G without isolated vertices, we show that the following inequality chains hold: s α ≤ β ≤ s β ≤ w β and s α ≤ w α ≤ α ≤ w β. Analogous to Gallai's theorem, we prove s β + w α = p and w β + s α = p. Further, we show that s α ≤ p - Δ and w α ≤ p - δ and find a necessary and sufficient condition to attain the upper bound, characterizing the graphs which attain these bounds. Several Nordhaus-Gaddum-type results and a Vizing-type result are also established.

Original language | English |
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Pages (from-to) | 1136-1145 |

Number of pages | 10 |

Journal | Discrete Mathematics |

Volume | 307 |

Issue number | 9-10 |

DOIs | |

Publication status | Published - 06-05-2007 |

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### All Science Journal Classification (ASJC) codes

- Discrete Mathematics and Combinatorics
- Theoretical Computer Science

### Cite this

*Discrete Mathematics*,

*307*(9-10), 1136-1145. https://doi.org/10.1016/j.disc.2006.07.040