# On Vertex-Weighted Graph Realizations

Amotz Bar-Noy, Toni Böhnlein, David Peleg, Dror Rawitz

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

## Abstract

Given a degree sequence d of length n, the Degree Realization problem is to decide if there exists a graph whose degree sequence is d, and if so, to construct one such graph. Consider the following natural variant of the 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 )wj= fi 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 the original degree realization problem, all the provided services must be equal to one. 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. In [7] it was observed that any requirements vector f where n is even can be realized. For odd n, the question of whether f is realizable is framed as whether fn (largest requirement) lies within certain intervals whose boundaries depend on the requirements f1, …, fn - 1. Intervals were identified where f can be realized but for their complements the question is left open. 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.

Original language English Algorithms and Complexity - 12th International Conference, CIAC 2021, Proceedings Tiziana Calamoneri, Federico Corò Springer Science and Business Media Deutschland GmbH 90-102 13 9783030752415 https://doi.org/10.1007/978-3-030-75242-2_6 Published - 2021 12th International Conference on Algorithms and Complexity, CIAC 2021 - Virtual, OnlineDuration: 10 May 2021 → 12 May 2021

### Publication series

Name Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) 12701 LNCS 0302-9743 1611-3349

### Conference

Conference 12th International Conference on Algorithms and Complexity, CIAC 2021 Virtual, Online 10/05/21 → 12/05/21