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

Amotz Bar-Noy, David Peleg, Dror Rawitz

Research output: Chapter in Book/Report/Conference proceedingConference contribution

## Abstract

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.
Original language English WALCOM: Algorithms and Computation Algorithms and Computation - 16th International Conference and Workshops, WALCOM 2022, Proceedings Petra Mutzel, Md. Saidur Rahman, Slamin Springer Science and Business Media Deutschland GmbH 315-327 13 9783030967314 3030967301, 9783030967307 https://doi.org/10.1007/978-3-030-96731-4_26 Published - 2022 16th International Conference and Workshops, WALCOM 2022 - Jember, IndonesiaDuration: 24 Mar 2022 → 26 Mar 2022

### Publication series

Name Lecture Notes in Computer Science 13174 0302-9743

### Conference

Conference 16th International Conference and Workshops, WALCOM 2022 24/03/22 → 26/03/22

## Keywords

• Degree sequence
• Graph realization
• Graph-algorithms

## All Science Journal Classification (ASJC) codes

• Theoretical Computer Science
• Computer Science(all)

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