Some results for Roman domination number on Cardinal product of paths and cycles
Aneta Klobucar, Ivona Puljić · Kragujevac Journal of Mathematics · 2014
For a graph G = (V, E), a Roman dominating function (RDF) is a function ƒ : V ( {0,1, 2} satisfying the condition that every vertex u for which ƒ(u) = 0 is adjacent to at least one vertex for which ƒ(v) = 2. The weight of an RDF equals w(f)=Σv=V F(v) = |V1| + 2|V2| where Vi = {v∊V : ƒ(v) = i}, i ∊{1, 2}. An RDF for which w(ƒ) achieves its minimum is called a γR-function and its weight, denoted by γR(G), is called the Roman domination number. In this paper we determine a lower and the upper bounds for γR(Pm x Pn) as well as the exact value of lim m,n→∞γR(Pm x Pn)/ mn where Pm x Pn stands for the cardinal product of two paths. We also present some results concerning the cardinal product of two cycles Cm x Cn as well as the exact value of lim m,n→∞γR(Cm x Cn)/ mn .