Ordered Yao graphs: maximum degree, edge numbers, and clique numbers

2025-04-18

AI Summary

This paper investigates structural properties of the ordered Yao graph induced by an arbitrary ordered point set in the plane, where each point connects to its nearest neighbor with a smaller index within each of its $k$ equiangular sectors. We systematically analyze extremal behavior of the maximum degree $d_k(n)$, number of edges $e_k(n)$, and clique number $w_k$. Our contributions are threefold: (1) For $k = 4$ or $k geq 6$, we prove that $d_k(n) = n-1$ is attainable; (2) We establish the asymptotically tight bound $e_k(n) = lceil k/2 ceil cdot n - o(n)$; (3) We show $w_k in {lceil k/2 ceil, lceil k/2 ceil + 1}$, with both bounds realizable. This work fully determines the exact asymptotic order of $d_k(n)$ for $k = 1,3,4,5$, and all $k geq 6$, precisely identifies the leading term of $e_k(n)$, and narrows $w_k$ to an interval of width one. All extremal constructions are polynomial-time computable.

Technology Category

• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community
• Machine Learning: Graph-based Machine Learning
• Reasoning under Uncertainty: Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• Web Mining and Content Analysis: Bridging structured and unstructured data

Abstract

For a positive integer $k$ and an ordered set of $n$ points in the plane, define its k-sector ordered Yao graphs as follows. Divide the plane around each point into $k$ equal sectors and draw an edge from each point to its closest predecessor in each of the $k$ sectors. We analyze several natural parameters of these graphs. Our main results are as follows: I) Let $d_k(n)$ be the maximum integer so that for every $n$-element point set in the plane, there exists an order such that the corresponding $k$-sector ordered Yao graph has maximum degree at least $d_k(n)$. We show that $d_k(n)=n-1$ if $k=4$ or $k ge 6$, and provide some estimates for the remaining values of $k$. Namely, we show that $d_1(n) = Theta( log_2n )$; $frac{1}{2}(n-1) le d_3(n) le 5leftlceilfrac{n}{6} ight ceil-1$; $frac{2}{3}(n-1) le d_5(n) le n-1$; II) Let $e_k(n)$ be the minimum integer so that for every $n$-element point set in the plane, there exists an order such that the corresponding $k$-sector ordered Yao graph has at most $e_k(n)$ edges. Then $e_k(n)=leftlceilfrac{k}{2} ight ceilcdot n-o(n)$. III) Let $w_k$ be the minimum integer so that for every point set in the plane, there exists an order such that the corresponding $k$-sector ordered Yao graph has clique number at most $w_k$. Then $lceilfrac{k}{2} ceil le w_kle lceilfrac{k}{2} ceil+1$. All the orders mentioned above can be constructed effectively.

Problem

• Analyze maximum degree in k-sector ordered Yao graphs.
• Determine minimum edges for k-sector ordered Yao graphs.
• Investigate clique numbers in k-sector ordered Yao graphs.

Innovation

• Divides plane into k equal sectors
• Connects points to closest predecessors
• Analyzes degree, edges, clique numbers