The influence of special vertices on strong domination

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′ = Ø we characterize all extremal graphs.