Vertex-edge perfect Roman domination number

A vertex-edge perfect Roman dominating function on a graph $G=(V,E)$ (denoted by ve-PRDF) is a function $f:V\left(G\right)⟶\{0,1,2\}$ such that for every edge $uv\in E$, $max\{f(u),f(v)\}\ne 0$, 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 $w(f)=\sum _{v\in V}f(v)$. 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.