On vertex-weighted realizations of acyclic and general graphs

Consider the following natural variation of the degree realization problem. Let G=(V,E) be a simple undirected graph of order n. Let f∈R≥0n be a vector of vertex requirements, and let w∈R≥0n be a vector of provided services at the vertices. Then w satisfies f on G if the constraints ∑j∈N(i)wj=fi are satisfied for all i∈V, where N(i) denotes the neighbourhood of vector i. Given a requirements vector f, the Vertex-Weighted Graph Realization problem asks for a suitable graph G and a vector w of provided services that satisfy f on G.