TY - GEN

T1 - Graph realizations

T2 - 17th Scandinavian Symposium and Workshops on Algorithm Theory, SWAT 2020

AU - Bar-Noy, Amotz

AU - Choudhary, Keerti

AU - Peleg, David

AU - Rawitz, Dror

N1 - Publisher Copyright:
© Amotz Bar-Noy, Keerti Choudhary, David Peleg, and Dror Rawitz; licensed under Creative Commons License CC-BY

PY - 2020/6/1

Y1 - 2020/6/1

N2 - The classical problem of degree sequence realizability asks whether or not a given sequence of n positive integers is equal to the degree sequence of some n-vertex undirected simple graph. While the realizability problem of degree sequences has been well studied for different classes of graphs, there has been relatively little work concerning the realizability of other types of information profiles, such as the vertex neighborhood profiles. In this paper, we initiate the study of neighborhood degree profiles, wherein, our focus is on the natural problem of realizing maximum neighborhood degrees. More specifically, we ask the following question: “Given a sequence D of n non-negative integers 0 ≤ d1 ≤ · · · ≤ dn, does there exist a simple graph with vertices v1, . . ., vn such that for every 1 ≤ i ≤ n, the maximum degree in the neighborhood of vi is exactly di?” We provide in this work various results for maximum-neighborhood-degree for general n vertex graphs. Our results are first of its kind that studies extremal neighborhood degree profiles. For closed as well as open neighborhood degree profiles, we provide a complete realizability criteria. We also provide tight bounds for the number of maximum neighbouring degree profiles of length n that are realizable. Our conditions are verifiable in linear time and our realizations can be constructed in polynomial time.

AB - The classical problem of degree sequence realizability asks whether or not a given sequence of n positive integers is equal to the degree sequence of some n-vertex undirected simple graph. While the realizability problem of degree sequences has been well studied for different classes of graphs, there has been relatively little work concerning the realizability of other types of information profiles, such as the vertex neighborhood profiles. In this paper, we initiate the study of neighborhood degree profiles, wherein, our focus is on the natural problem of realizing maximum neighborhood degrees. More specifically, we ask the following question: “Given a sequence D of n non-negative integers 0 ≤ d1 ≤ · · · ≤ dn, does there exist a simple graph with vertices v1, . . ., vn such that for every 1 ≤ i ≤ n, the maximum degree in the neighborhood of vi is exactly di?” We provide in this work various results for maximum-neighborhood-degree for general n vertex graphs. Our results are first of its kind that studies extremal neighborhood degree profiles. For closed as well as open neighborhood degree profiles, we provide a complete realizability criteria. We also provide tight bounds for the number of maximum neighbouring degree profiles of length n that are realizable. Our conditions are verifiable in linear time and our realizations can be constructed in polynomial time.

KW - Extremum-degree

KW - Graph realization

KW - Neighborhood profile

UR - http://www.scopus.com/inward/record.url?scp=85090394670&partnerID=8YFLogxK

U2 - 10.4230/LIPIcs.SWAT.2020.10

DO - 10.4230/LIPIcs.SWAT.2020.10

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AN - SCOPUS:85090394670

T3 - Leibniz International Proceedings in Informatics, LIPIcs

BT - 17th Scandinavian Symposium and Workshops on Algorithm Theory, SWAT 2020

A2 - Albers, Susanne

PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing

Y2 - 22 June 2020 through 24 June 2020

ER -