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

TL;DR

This work studies ordered $k$-sector Yao graphs on planar point sets, focusing on three natural metrics: the maximum indegree $d_k(n)$, the minimum guaranteed edge count $e_k(n)$, and the clique number $w_k$. It develops constructive orderings (orthogonal and radial) to maximize indegree, a geometric partition approach to bound edge counts, and complex-root constructions to bound clique sizes, yielding tight or near-tight bounds across a range of $k$ and $n$. The results include $d_k(n)=n-1$ for even $k$ or $k obreak\ge6$, $d_1(n)= obreak heta( obreak ),$ and bounds for $d_3(n)$ and $d_5(n)$; asymptotically, $e_k(n)=iglraceil rac{k}{2} igr ceil n - o(n)$ with refined estimates, and $iglraceil rac{k}{2} igrceil obreak\le w_k obreak obreak ext{(between this and one more)}$; all orders are effectively constructible. These findings advance understanding of deterministic ordering in geometric graphs, with implications for spanner-type properties and network design in computational geometry.

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 $d_1(n) = Θ( \log_2n )$; $\frac{1}{2}(n-1) \le d_3(n) \le 5\left\lceil\frac{n}{6}\right\rceil-1$; $\frac{2}{3}(n-1) \le d_5(n) \le n-1$; 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 $e_k(n)=\left\lceil\frac{k}{2}\right\rceil\cdot n-o(n)$. 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 $\lceil\frac{k}{2}\rceil \le w_k\le \lceil\frac{k}{2}\rceil+1$. All the orders mentioned above can be constructed effectively.

Figures (10)

• Figure 1: Unordered (left) and ordered (right) 3-sector Yao graphs on the same set of six points.
• Figure 2: $p$ lies in the first sector of $q$ if and only if $q$ lies in the first dual sector of $p$. Or to be short, $p\in s_0\left(q\right)$ and $q\in-s_0\left(q\right)$.
• Figure 3: Five dual sectors of $p$; the first two are empty.
• Figure 4: Both $c_m$ and $a_i$ belong to the first sector of $f_1$, both $f_1$ and $a_i$ belong to the third sector of $c_m$, $c_mf_1$ is the shortest side of the triangle $a_ic_mf_1$.
• Figure 5: A set with a triangle-free $3$-sector ordered Yao graph.

Theorems & Definitions (19)

• Definition 1
• Theorem 1
• Definition 2
• Theorem 2
• Definition 3
• Theorem 3
• Definition 4
• Theorem 4
• Definition 5
• Theorem 5