Vertex-Weighted Graphs: Realizable and Unrealizable Domains: Realizable and Unrealizable Domains

Consider the following natural variation of the degree realization problem. Let G=(V,E)G=(V,E) be a simple undirected graph of order n. Let f∈Rn≥0f∈R≥0n be a vector of vertex requirements, and let w∈Rn≥0w∈R≥0n be a vector of provided services at the vertices. Then w satisfies f on G if the constraints ∑j∈N(i)wj=fi∑j∈N(i)wj=fi are satisfied for all i∈Vi∈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 fnfn (the largest requirement) lies within certain intervals. In [5] some intervals are identified where f can be realized, and their complements form n−32n−32 connected intervals (“unknown domains”) which we give odd indices k=1,3,…,n−4k=1,3,…,n−4. The unknown domain for k=1k=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≥3k≥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.