Vertex-edge perfect Roman domination number

A vertex-edge perfect Roman dominating function on a graph G = (V, E) (denoted by vePRDF) is a function f : V (G) -& RARR; {0, 1, 2} such that for every edge uv & ISIN; E, max{ f(u), f(v)} # 0, 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) = Zv & ISIN;V f (v). The vertex-edge perfect Roman domination number of G (denoted by & gamma;pveR(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;pveR(T) in terms of the order n, l leaves and s support vertices. Lastly, we determine & gamma;pveR(G) for Petersen, cycle and Flower snark graphs.