Some Results on the Strong Roman Domination Number of Graphs

Let G=(V,E) be a finite and simple graph of order n and maximum‎ ‎degree Δ(G)‎. ‎A strong Roman dominating function on a‎ ‎graph G is a function f‎:V (G)→{0‎, ‎1,… ,‎[Δ(G)/2 ]‎+ ‎1} satisfying the condition that every‎ ‎vertex v for which f(v)=0 is adjacent to at least one vertex u ‎for which‎ f(u) ≤ 1‎+ [(1/2)| N(u) ∩ V0| ], ‎where V0={v ∊ V | f(v)=0}. The minimum of the‎ values ∑v∊ V f(v), ‎taken over all strong Roman dominating‎ ‎functions f of G‎, ‎is called the strong Roman domination‎ ‎number of G and is denoted by γStR(G)‎. ‎In this paper we‎ ‎continue the study of strong Roman domination number in graphs‎. ‎In‎ particular‎, ‎we present some sharp bounds for γStR(G) and‎ we determine the strong Roman domination number of some graphs‎.