Graph Realizations: Maximum and Minimum Degree in Vertex Neighborhoods

TL;DR

This paper studies the realizability of maximum and minimum neighborhood degree profiles in graphs, providing a complete characterization for maximum profiles and near-complete conditions for minimum profiles, advancing understanding of neighborhood degree constraints.

Contribution

It introduces the first comprehensive criteria for realizing extremal neighborhood degree profiles in graphs, especially focusing on maximum and minimum neighborhood degrees.

Findings

Complete realizability criteria for maximum neighborhood degree profiles.

Necessary and sufficient conditions for minimum profiles differ by at most a factor of two.

First study of extremal neighborhood degree profiles in graph realizability.

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. 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 $0\leq d_1\leq \cdots \leq d_n$, does there exist a simple graph with vertices $v_1,\ldots, v_n$ such that for every $1\le i \le n$, the maximum (resp. minimum) degree in the neighborhood of $v_i$ is exactly $d_i$? 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}.