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)w_j=f_i 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. 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 which was started in [2]. For the unsolved cases, the question of whether a vector f is realizable can be formulated as whether its largest requirement lies within certain intervals. 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.