The Perfect Roman Domination Number of the Cartesian Product of Some Graphs
Ahlam Almulhim, Abolape Deborah Akwu, Bana Al Subaiei · Journal of Mathematics · 2022
A perfect Roman dominating function on a graphGis a functionf:V(G)⟶{0,1,2} for which every vertexvwithf(v) = 0 is adjacent to exactly one neighboruwithf(u) = 2. The weight offis the sum of the weights of the vertices. The perfect Roman domination number of a graphG, denoted by , is the minimum weight of a perfect Roman dominating function onG. In this paper, we prove that ifGis the Cartesian product of a pathPrand a pathPs, a pathPrand a cycleCs, or a cycleCrand a cycleCs, wherer,s> 5, then .