Vertex-Weighted Graphs: Realizable and Unrealizable Domains

Abstract

Consider the following natural variation of the degree realization problem. Let $G=(V, E)$ be a simple undirected graph of order n. Let $f \in \mathbb {R}_{\ge 0}^{n}$ be a vector of vertex requirements, and let $w\in \mathbb {R}_{\ge 0}^{n}$ be a vector of provided services at the vertices. Then w satisfies f on G if the constraints $\sum _{j \in N(i)} w_j = f_i$ are satisfied for all $i \in V$, 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 $\frac{n-3}{2}$ connected intervals (“unknown domains”) which we give odd indices $k = 1,3,\ldots , n-4$. 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 $k \ge 3$. 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.

Supported in part by a US-Israel BSF grant (2018043). Partly supported by ARL Cooperative Grant, ARL Network Science CTA, W911NF-09-2-0053.