The Perfect Roman Domination Number of the Cartesian Product of Some Graphs

Abstract

A perfect Roman dominating function on a graph G is a function f : V G ⟶ 0,1,2 for which every vertex v with f v = 0 is adjacent to exactly one neighbor u with f u = 2 . The weight of f is the sum of the weights of the vertices. The perfect Roman domination number of a graph G , denoted by γ R p G , is the minimum weight of a perfect Roman dominating function on G . In this paper, we prove that if G is the Cartesian product of a path P r and a path P s , a path P r and a cycle C s , or a cycle C r and a cycle C s , where r , s > 5 , then γ R p G ≤ 2 / 3 G .

Total Perfect Roman Domination

“…Since then, over 100 papers on Roman domination and its variants have been published. Perfect Roman domination [5,6], Italian domination [7,8], perfect Italian domination [9], double-Roman domination [10], perfect double-Roman domination [11], double-Italian domination [12], total Roman domination [13,14], vertex-edge Roman domination [15], and vertex-edge perfect Roman domination [16] are a few examples of Roman domination variants.…”