On strong (weak) independent sets and vertex coverings of a graph

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.