Vertex-Weighted Graphs: Realizable and Unrealizable Domains

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 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 [7] it is observed that any requirement vector where n is even can be realized. If n is odd, the problem becomes much harder. For the unsolved cases, the decision of whether f is realizable or not can be formulated as whether f_n (the largest requirement) lies within certain intervals. In [5] some intervals are identified where f can be realized, and their complements form n-32 connected intervals (“unknown domains”) which we give odd indices k= 1, 3, …, n- 4. The unknown domain for k= 1 is shown to be unrealizable. Our main result presents structural properties that a graph must have if it realizes a vector in one of these unknown domains for k≥ 3. The unknown domains are characterized by inequalities which we translate to graph properties. Our analysis identifies several realizable sub-intervals, and shows that each of the unknown domains has at least one sub-interval that cannot be realized.