2019/12/31 by Amotz Bar-Noy, Keerti Choudhary, Bar-Noy, Amotz +5
Computer Science · #Advanced Graph Theory Research #Complexity and Algorithms in Graphs #Data Structures and Algorithms (cs.DS) #Digital Image Processing Techniques #Discrete Mathematics (cs.DM) #FOS: Computer and information sciences
paper · pdf · doi:10.48550/arxiv.1912.13286
openalex publication_date 2019/12/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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. We focus on the natural problem of realizing maximum and minimum 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 (resp. minimum) degree in the neighborhood of vi is exactly di? We provide in this work various results for both maximum as well as minimum neighborhood degree for general n vertex graphs. Our results are first of its kind that studies extremal neighborhood degree profiles. For maximum neighborhood degree profiles, we provide a \em complete realizability criteria. In comparison, we observe that the minimum neighborhood profiles are not so well-behaved, for these our necessary and sufficient conditions for realizability \em differ by a factor of at most two.