TY - GEN

T1 - Vertex-Weighted Graphs: Realizable and Unrealizable Domains

T2 - 16th International Conference and Workshops, WALCOM 2022

AU - Bar-Noy, Amotz

AU - Peleg, David

AU - Rawitz, Dror

N1 - Publisher Copyright: © 2022, Springer Nature Switzerland AG.

PY - 2022

Y1 - 2022

N2 - 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.

AB - 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.

KW - Degree sequence

KW - Graph realization

KW - Graph-algorithms

UR - http://www.scopus.com/inward/record.url?scp=85127851283&partnerID=8YFLogxK

U2 - https://doi.org/10.1007/978-3-030-96731-4_26

DO - https://doi.org/10.1007/978-3-030-96731-4_26

M3 - منشور من مؤتمر

SN - 3030967301

SN - 9783030967307

T3 - Lecture Notes in Computer Science

SP - 315

EP - 327

BT - WALCOM: Algorithms and Computation

A2 - Mutzel, Petra

A2 - Rahman, Md. Saidur

A2 - Slamin, null

PB - Springer Science and Business Media Deutschland GmbH

Y2 - 24 March 2022 through 26 March 2022

ER -