Graph Realizations: Maximum Degree in Vertex Neighborhoods
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.
Graph realization
neighborhood profile
extremum-degree
Mathematics of computing~Graph theory
10:1-10:17
Regular Paper
W911NF-09-2-0053 (the ARL Network Science CTA), US-Israel BSF grant 2018043.
Amotz
Bar-Noy
Amotz Bar-Noy
City University of New York (CUNY), NY, USA
Keerti
Choudhary
Keerti Choudhary
Tel Aviv University, Israel
David
Peleg
David Peleg
Weizmann Institute of Science, Rehovot, Israel
Dror
Rawitz
Dror Rawitz
Bar Ilan University, Ramat-Gan, Israel
10.4230/LIPIcs.SWAT.2020.10
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Amotz Bar-Noy, Keerti Choudhary, David Peleg, and Dror Rawitz
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