On Vertex-Weighted Graph Realizations

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

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


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 languageEnglish
Title of host publicationAlgorithms and Complexity - 12th International Conference, CIAC 2021, Proceedings
EditorsTiziana Calamoneri, Federico Corò
PublisherSpringer Science and Business Media Deutschland GmbH
Number of pages13
ISBN (Print)9783030752415
StatePublished - 2021
Event12th International Conference on Algorithms and Complexity, CIAC 2021 - Virtual, Online
Duration: 10 May 202112 May 2021

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume12701 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349


Conference12th International Conference on Algorithms and Complexity, CIAC 2021
CityVirtual, Online

Bibliographical note

Publisher Copyright:
© 2021, Springer Nature Switzerland AG.


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