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Graph Realizations: Maximum Degree in Vertex Neighborhoods

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



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Amotz Bar-Noy
  • City University of New York (CUNY), NY, USA
Keerti Choudhary
  • Tel Aviv University, Israel
David Peleg
  • Weizmann Institute of Science, Rehovot, Israel
Dror Rawitz
  • Bar Ilan University, Ramat-Gan, Israel

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Amotz Bar-Noy, Keerti Choudhary, David Peleg, and Dror Rawitz. Graph Realizations: Maximum Degree in Vertex Neighborhoods. In 17th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 162, pp. 10:1-10:17, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)
https://doi.org/10.4230/LIPIcs.SWAT.2020.10

Abstract

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≤ ⋯ ≤ d_n, does there exist a simple graph with vertices v₁,…, v_n such that for every 1≤ i ≤ n, the maximum degree in the neighborhood of v_i is exactly d_i?" 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.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Graph theory
Keywords
  • Graph realization
  • neighborhood profile
  • extremum-degree

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