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, namely, an n-vertex graph whose degree sequence is d¯, and if so, to construct one such realization. The problem was well researched over the recent decades and plays an important role in the field of Social Networks. In this paper, 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¯∈R₊ ⁿ be a vector of requirements of the vertices, and let w¯∈R₊ ⁿ be a vector of provided services at the vertices. The provided services vector w¯ satisfies the requirements vector f¯ on G if the constraints ∑_j∈Γ(i)w_j=f_i are satisfied for all i∈V, where Γ(i) denotes the neighborhood of i. We study the following weighted graph realization problem. Given a requirements vector f¯, the goal is to find a suitable graph G and a vector w¯ of provided services that satisfy f¯ on G. In the original degree realization problem, all the provided services must be equal to one. For even n, we show that every requirement vector is realizable. For odd n, the picture is more complicated, as certain requirement vectors are non-realizable. We provide a complete characterization for n=3 and n=5, and give (non-matching but close) necessary and sufficient conditions for realizability for odd n≥7. We provide a complete characterization for the variant in which the constraints that should be satisfied are: max_j∈Γ(i)⁡w_j=f_i, for all i∈V. As before, we show that every requirement vector can be realized if n is even. For odd n, we show that a vector is realizable if and only if not all requirements are distinct.