TY - JOUR

T1 - On vertex-weighted realizations of acyclic and general graphs

AU - Bar-Noy, Amotz

AU - Böhnlein, Toni

AU - Peleg, David

AU - Rawitz, Dror

N1 - Publisher Copyright:
© 2022 Elsevier B.V.

PY - 2022/6/24

Y1 - 2022/6/24

N2 - Consider the following natural variation of the degree realization 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 neighbourhood of vector i. Given a requirements vector f, the VERTEX-WEIGHTED GRAPH REALIZATION problem asks for a suitable graph G and a vector w of provided services that satisfy f on G. 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 which was started in [2]. For the unsolved cases, the question of whether a vector f is realizable can be formulated as whether its largest requirement lies within certain intervals. 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.

AB - Consider the following natural variation of the degree realization 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 neighbourhood of vector i. Given a requirements vector f, the VERTEX-WEIGHTED GRAPH REALIZATION problem asks for a suitable graph G and a vector w of provided services that satisfy f on G. 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 which was started in [2]. For the unsolved cases, the question of whether a vector f is realizable can be formulated as whether its largest requirement lies within certain intervals. 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.

KW - Degree sequences

KW - Graph algorithms

KW - Graph realization

UR - http://www.scopus.com/inward/record.url?scp=85128786468&partnerID=8YFLogxK

U2 - 10.1016/j.tcs.2022.04.012

DO - 10.1016/j.tcs.2022.04.012

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AN - SCOPUS:85128786468

SN - 0304-3975

VL - 922

SP - 81

EP - 95

JO - Theoretical Computer Science

JF - Theoretical Computer Science

ER -