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- W2000920682 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 number s 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 number s 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." @default.
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- W2000920682 date "2007-05-01" @default.
- W2000920682 modified "2023-09-24" @default.
- W2000920682 title "On strong (weak) independent sets and vertex coverings of a graph" @default.
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- W2000920682 doi "https://doi.org/10.1016/j.disc.2006.07.040" @default.
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