Vertex-weighted realizations of graphs

Given a degree sequence d of length n, the degree realization problem is to decide if d has a realization. That is a n-vertex graph whose degree sequence is d, and if this is the case, to construct such a realization (cf. [6, 7, 8]). We consider the following natural generalization of the problem: Let G = (V,E) be a simple undirected graph on V = {1, 2, . . . , n}. Let f 2 Nn be a vector of vertex-requirements, and let w 2 Nn be a vector of vertex-weights. The weight vector w satisfies the requirement vector f on G if the constraints P j2..(i) wj = fi are satisfied for all i 2 V , where ..(i) denotes the neighborhood of i. The vertex-weighted realization problem is now as follows: Given a requirements vector f, find a suitable graph G and a weight vector w that satisfy f on G. In the original degree realization problem, all vertex weights are equal to one.