TY - JOUR

T1 - Graph realizations

T2 - Maximum degree in vertex neighborhoods

AU - Bar-Noy, Amotz

AU - Choudhary, Keerti

AU - Peleg, David

AU - Rawitz, Dror

N1 - Publisher Copyright:
© 2023

PY - 2023/9

Y1 - 2023/9

N2 - This paper initiates the study of the maximum neighborhood degree realization problem. Given a sequence D=(d1,…,dn) of non-negative integers, the goal is to construct a simple graph with vertices v1,…,vn such that for every i∈[1,n], the maximum degree in the neighborhood of vi is exactly di (or output null if no such graph exists). Depending upon whether or not the realizing graph is required to be connected, and whether or not the neighborhood of a vertex is closed (that is, the neighborhood includes the vertex itself), the problem has four natural settings. We provide complete realizability criteria for all four settings of the problem. Our conditions are verifiable in linear time and our realizations can be constructed in polynomial time. In addition, we prove tight/approximate bounds for the number of maximum neighboring degree profiles of length n that are realizable.

AB - This paper initiates the study of the maximum neighborhood degree realization problem. Given a sequence D=(d1,…,dn) of non-negative integers, the goal is to construct a simple graph with vertices v1,…,vn such that for every i∈[1,n], the maximum degree in the neighborhood of vi is exactly di (or output null if no such graph exists). Depending upon whether or not the realizing graph is required to be connected, and whether or not the neighborhood of a vertex is closed (that is, the neighborhood includes the vertex itself), the problem has four natural settings. We provide complete realizability criteria for all four settings of the problem. Our conditions are verifiable in linear time and our realizations can be constructed in polynomial time. In addition, we prove tight/approximate bounds for the number of maximum neighboring degree profiles of length n that are realizable.

KW - Graph realization

KW - Maximum neighborhood degree

KW - Neighborhood degree profile

UR - http://www.scopus.com/inward/record.url?scp=85159837069&partnerID=8YFLogxK

U2 - 10.1016/j.disc.2023.113483

DO - 10.1016/j.disc.2023.113483

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AN - SCOPUS:85159837069

SN - 0012-365X

VL - 346

JO - Discrete Mathematics

JF - Discrete Mathematics

IS - 9

M1 - 113483

ER -