On Vertex-Weighted Graph Realizations

Given a degree sequence d of length n, the Degree Realization problem is to decide if there exists a graph whose degree sequence is d, and if so, to construct one such graph. Consider the following natural variant of the 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 )}w_j= f_i are satisfied for all i∈ V, where N(i) denotes the neighborhood of i. Given a requirements vector f, the Weighted Graph Realization problem asks for 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. In this paper, we consider two avenues. We initiate a study that focuses on weighted realizations where the graph is required to be of a specific class by providing a full characterization of realizable requirement vectors for paths and acyclic graphs. However, checking the respective criteria is shown to be NP-hard. In the second part, we advance the study in general graphs. In [7] it was observed that any requirements vector f where n is even can be realized. For odd n, the question of whether f is realizable is framed as whether f_n (largest requirement) lies within certain intervals whose boundaries depend on the requirements f₁, …, f_{n - 1}. Intervals were identified where f can be realized but for their complements the question is left open. We describe several new, realizable intervals and show the existence of an interval that cannot be realized. The complete classification for general graphs is an open problem.