[논문 리뷰] On the Largest Convexity Number of Co-Finite Sets in the Plane

Chaya Keller, Micha A. Perles|arXiv (Cornell University)|2026. 01. 01.

Computational Geometry and Mesh Generation인용 수 0

한 줄 요약

본 논문은 평면에서 n개의 점을 뺀 경우의 볼록성 수 gamma에 대한 예리하고 근사 예리한 경계를 결정하며, 볼록 위치에서의 정확한 값을 자세히 제시하고 일반 위치에 대해서는 상한을 제공한다. 여기에는 분리-커버링 변형도 포함한다.

ABSTRACT

The convexity number of a set $X \subset \mathbb{R}^2$ is the minimum number of convex subsets required to cover it. We study the following question: what is the largest possible convexity number $f(n)$ of $\mathbb{R}^2 \setminus S$, where $S$ is a set of $n$ points in general position in the plane? We prove that for all $n \geq 4$, $\lfloor\frac{n+5}{2} floor \leq f(n) \leq \frac{7n+44}{11}$. We also show that for every $n \geq 4$, if the points of $S$ are in convex position then the convexity number of $\mathbb{R}^2 \setminus S$ is $\lfloor\frac{n+5}{2} floor$. This solves a problem of Lawrence and Morris [Finite sets as complements of finite unions of convex sets, Disc. Comput. Geom. 42 (2009), 206-218].

연구 동기 및 목표

• Motivate and quantify how many convex sets are needed to cover or encapsulate the plane minus a finite point set in the plane.
• Characterize gamma(X) for X = R^2 \backslash P with P of size n in both convex and general position, including disjoint-cover variants.
• Resolve questions posed by Lawrence and Morris on co-finite sets and convex coverings in the plane.
• Provide precise bounds linking the size of P to covering/encapsulation counts and analyze convex-position versus general-position scenarios.

제안 방법

• Define covering and encapsulation parameters for R^2 \backslash P under several variants (cover, cover without intersection, encapsulate, encapsulate disjoint).
• Derive lower bounds for encapsulation in convex position using geometric arguments and case analysis (including a base for n=4 and inductive steps).
• Prove upper bounds by constructive coverings using half-planes and intersections, and through inductive decomposition for general P.

실험 결과

연구 질문

• RQ1 What is the maximum convexity number f(n) = max_P:|P|=n gamma(R^2 \backslash P) for P in general position?
• RQ2 What are the exact values of enc(P) and cov(P) when P is in convex position, and how do these relate to disjoint variants enc_{\nobreak}{\u0000a0}^{\nobreak c} and cov_{\nobreak}{\u0000a0}^{\nobreak c}?
• RQ3 How do the convexity bounds change when considering disjoint coverings or encapsulation of convex hulls?
• RQ4 Do the convex-position results resolve the problem posed by Lawrence and Morris for the convex and disjoint settings?
• RQ5 Can the results in the plane be extended to relate gamma(X) to other combinatorial or chromatic parameters of the invisibility graph?

주요 결과

• For n points in convex position, enc(P)=cov(P)= floor((n+5)/2) - delta(n), with delta(n)=1 for n in {0,1,3} and 0 otherwise.
• For general position, enc(P) and cov(P) are bounded above by 7n/11 + 4.
• In the disjoint (enc_{\u0000a0}^{\u0000a0c}, cov_{\u0000a0}^{\u0000a0c}) setting, enc_{\u0000a0}^{\u0000a0c}(n)=cov_{\u0000a0}^{\u0000a0c}(n)= floor((2n+5)/3).
• The results fully resolve the disjoint and convex settings of the problem for general position in the plane, paralleling and completing prior questions from Lawrence and Morris.
• The paper confirms that enc(n) and cov(n) remain open with a significant gap between the proven lower and upper bounds, marking a key direction for future research.
• The exact convex-position results support and settle the convex-variant questions, providing tight expressions for both encapsulation and covering counts.