Graph realizations: Maximum degree in vertex neighborhoods

Amotz Bar-Noy, Keerti Choudhary, David Peleg, Dror Rawitz

Research output: Contribution to journalArticlepeer-review


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 & ISIN; [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.& COPY; 2023 Published by Elsevier B.V.
Original languageEnglish
Article number113483
Number of pages16
JournalDiscrete Mathematics
Issue number9
Early online date23 May 2023
StatePublished - Sep 2023


  • Graph realization
  • Maximum neighborhood degree
  • Neighborhood degree profile

All Science Journal Classification (ASJC) codes

  • Theoretical Computer Science
  • Discrete Mathematics and Combinatorics


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