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 .

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<abstract><p>A vertex-edge perfect Roman dominating function on a graph $ G = (V, E) $ (denoted by ve-PRDF) is a function $ f:V\left(G\right)\longrightarrow\{0, 1, 2\} $ such that for every edge $ uv\in E $, $ \max\{f(u), f(v)\}\neq0 $, or $ u $ is adjacent to exactly one neighbor $ w $ such that $ f(w) = 2 $, or $ v $ is adjacent to exactly one neighbor $ w $ such that $ f(w) = 2 $. The weight of a ve-PRDF on $ G $ is the sum $ w(f) = \sum_{v\in V}f(v) $. The vertex-edge perfect Roman domination number of $ G $ (denoted by $ \gamma_{veR}^{p}(G) $) is the minimum weight of a ve-PRDF on $ G $. In this paper, we first show that vertex-edge perfect Roman dominating is NP-complete for bipartite graphs. Also, for a tree $ T $, we give upper and lower bounds for $ \gamma_{veR}^{p}(T) $ in terms of the order $ n $, $ l $ leaves and $ s $ support vertices. Lastly, we determine $ \gamma_{veR}^{p}(G) $ for Petersen, cycle and Flower snark graphs.</p></abstract>