Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 1410.0309

10-Gabriel graphs are Hamiltonian

Tomáš Kaiser·Maria Saumell·Nico Van Cleemput
View original source ↗Read source document ↗Export metadata ↓

Abstract

Given a set $S$ of points in the plane, the $k$-Gabriel graph of $S$ is the geometric graph with vertex set $S$, where $p_i,p_j\in S$ are connected by an edge if and only if the closed disk having segment $\bar{p_ip_j}$ as diameter contains at most $k$ points of $S \setminus \{p_i,p_j\}$. We consider the following question: What is the minimum value of $k$ such that the $k$-Gabriel graph of every point set $S$ contains a Hamiltonian cycle? For this value, we give an upper bound of 10 and a lower bound of 2. The best previously known values were 15 and 1, respectively.

Explore related subjects

cs.CG

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Tomáš Kaiser, Maria Saumell, Nico Van Cleemput. 2015-06-06. 10-Gabriel graphs are Hamiltonian. https://arxiv.org/abs/1410.0309

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Unlabeled Multi-Robot Motion Planning with Improved Separation Trade-offs

We study unlabeled MRMP for unit-disk robots in a polygonal environment. Although the problem is hard in general, polynomial-time solutions exist under appropriate separation assumptions on start and target positions. Banyassady et al.(SoCG'22) guarantee feasibility in simple polygons under start--start and target--target distances of at least $4$, and start--target distances of at least $3$, but without optimality guarantees. Solovey et al.(RSS'15) provide a near-optimal solution in general polygonal domains, under stricter conditions: start/target positions must have pairwise distance at least $4$, and at least $\sqrt{5}\approx2.236$ from obstacles. This raises the question of whether polynomial-time algorithms can be obtained in even more densely packed environments. In this paper we present a generalized algorithm that achieve different tradeoffs on the robots-separation $ρ$ and obstacles-separation $ω$, all significantly improving upon the state of the art. Specifically, we obtain polynomial-time constant-approximation algorithms to minimize the total path length when (i) $ρ=2\frac{2}{3}$ and $ω=1\frac{2}{3}$, or (ii) $ρ\approx3.291$ and $ω\approx1.354$. These solutions are weakly-monotone; we also provide a monotone solution requiring $ω=\approx1.614$ and $ρ=4$. We prove that monotone plans may not exist when $ω<1.614$, and weakly-monotone plans may not exist when $ω<1.354$. We then present tradeoffs between the separation bounds and the approximation factor, specifically achieving an (almost) optimal bound of $ρ=2$ at the cost of a linear approximation factor and requiring $ω=2$. This applies also for the labeled variant of MRMP, in which case we show a tight bound on $ω$. Finally, we show that without any robots-separation assumption, obstacles-separation of at least $1.5$ may be necessary for a solution to exist.

cs.CG↗
arXiv2026

Low-Dimensional Embeddings for Gaussian Kernels on Manifolds

The Gaussian kernel is a widely used similarity measure underlying kernel methods such as kernel PCA and spectral clustering, but computing Gaussian kernel distances for many pairs of points can be expensive. Using Random Fourier Features (RFF), Chen and Phillips [ALT 2017] showed that for points in a $d$-dimensional Euclidean ball in ${\mathbb R}^N$, $t=Ω((d/\varepsilon^2)\log(dR/\varepsilon))$ features suffice to preserve all pairwise Gaussian kernel distances within a $(1\pm\varepsilon)$ factor with high probability. We establish a uniform relative-error embedding theorem for the more general setting of an arbitrary positive-reach submanifold $\mathcal M\subset{\mathbb R}^N$ of intrinsic dimension $d$. We show that $t=O((d/\varepsilon^2)\log(\operatorname{vol}(\mathcal M)^2N^{2d}/(\operatorname{vol}(B_1^d(0))^2\operatorname{rch}(\mathcal M)^{2d}\varepsilon^{2d+1}δ)))$, or approximately $O((d^2/\varepsilon^2)(\log N+\log(1/(\varepsilonδ))))$, RFFs suffice, with probability $1-δ$, to preserve the Gaussian kernel distance between every pair of manifold points up to relative error $\varepsilon$. Thus the bound depends only logarithmically on the ambient dimension and on manifold parameters such as volume and reach, while retaining the $1/\varepsilon^2$ Euclidean rate. We also prove a topological consequence: under the same RFF embedding, persistent homology is preserved in the sense that weighted Cech and Rips filtrations built from Gaussian kernel power distance are $(1\pm\varepsilon_\star)$-interleaved, where $\varepsilon_\star$ accounts for both distance distortion and kernel-weight approximation.

cs.CG↗
arXiv2026

Perfectly Guarding Straits: Exact Algorithms for Weak Visibility Polygons

The Art Gallery Problem (AGP) asks for the fewest guards that see all of a simple polygon. It is $\exists\mathbb{R}$-complete, hence NP-hard. We show that for a particular class of polygons, confining guards to a single edge makes AGP exactly and efficiently solvable. We call this the Strait Guarding Problem (SGP). Its input is a weak visibility polygon (WVP): a simple polygon where every point is seen from some point of one fixed edge, the base. SGP places the fewest guards on the base that jointly see the whole polygon. First, a structural fact: guards on the base edge that cover the boundary already cover the entire interior, turning a two-dimensional covering problem into a one-dimensional one. Our main result is the Witness-Guard Algorithm, which solves SGP exactly in $O((n + \mathrm{OPT} \cdot ρ)(\log n + \log \mathrm{OPT}))$ time, where $ρ$ is the number of reflex vertices in the WVP and OPT is the minimum number of guards. It is output-sensitive and certifies optimality by a witness set of size OPT derived from its output. We also study the guarding-the-vertex version and prove a tight $Θ(n \log n)$ bound, with the lower bound following from Sorting. As a corollary of SGP, we obtain two results for altitude terrain guarding (ATG), a special case that SGP generalizes. We give a linear-time perfect-guarding algorithm, improving the previous $O(n^2 \log n)$ bound of Daescu, Friedrichs, Malik, Polishchuk and Schmidt. We also resolve their problem on the minimum guarding altitude, in $O(nk + k^2 \log k)$ time, improving on the $O(k^2 λ_{k-1}(n) \log n)$ bound of Kang, Kim and Ahn.

cs.CG↗