Vertex-weighted realizations of graphs

Given a degree sequence (d) over bar of length n, the degree realization problem is to decide if (d) over bar has a realization, namely, an n-vertex graph whose degree sequence is (d) over bar, 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) over bar is an element of R-+(n) be a vector of requirements of the vertices, and let (w) over bar is an element of R-+(n) be a vector of provided services at the vertices. The provided services vector (w) over bar satisfies the requirements vector (f) over bar on G if the constraints Sigma(j is an element of Gamma(i)) w(j) = f(i) are satisfied for all i is an element of V, where Gamma(i) denotes the neighborhood of i. We study the following weighted graph realization problem. Given a requirements vector (f) over bar, the goal is to find a suitable graph G and a vector (w) over bar of provided services that satisfy (f) over bar 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 is an element of Gamma(i)) w(j) = f(i), for all i is an element of 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. (C) 2019 Elsevier B.V. All rights reserved.