Reduction to a vertex cover problem-like with weighted vertices and edges
This problem is a generalization of Dominating Set, and if you let weights of vertices and edges be 1, then you have exactly Dominating Set.

The problem definition of Dominating Set is:


  Given a graph $G = (V, E)$ and an integer $k$, is there a set $V' \subseteq V$ of size at most $k$ such that the distance of any vertex $v \in V \setminus V'$  to $V'$ is 1.


(This is assuming that the distance from $v' \in V'$ to $V'$ is defined as 0.)

It follows that WVE is NP-hard.  This is occasionally called proof by generalization.