Abstract
Abstract Let G = ( V , E ) be a graph. A set D ⊆ V is a strong dominating set of G if for every vertex y ∈ V − D there is a vertex x ∈ D with xy ∈ E of larger or equal degree, i.e. d ( y , G ) ⩽ d ( x , G ). The strong domination number γ st ( G ) is defined as the minimum cardinality of a strong dominating set and was introduced by Sampathkumar and Pushpa Latha in 1996. Let I be the set of vertices of G without neighbours of larger or equal degree. It is known that γ st ( G ) ⩽ (| V | + | I |)/2. We show that the influence of | I | on γ st ( G ) is actually weaker. We present a new bound on γ st ( G ) where | I | in the above expression is replaced by max {1, | I ′|} for a suitable subset I ′ of I . In the special case I ′ = O we characterize all extremal graphs.
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