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Hamiltonicity for Convex Shape Delaunay and Gabriel Graphs
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SYSNO ASEP 0523630 Document Type J - Journal Article R&D Document Type Journal Article Subsidiary J Článek ve WOS Title Hamiltonicity for Convex Shape Delaunay and Gabriel Graphs Author(s) Bose, P. (CA)
Cano, P. (CA)
Saumell, Maria (UIVT-O) RID, SAI, ORCID
Silveira, R.I. (ES)Article number 101629 Source Title Computational Geometry-Theory and Applications. - : Elsevier - ISSN 0925-7721
Roč. 89, August 2020 (2020)Number of pages 17 s. Publication form Online - E Language eng - English Country NL - Netherlands Keywords Delaunay graphs ; Hamiltonicity ; Gabriel graphs Subject RIV BA - General Mathematics OECD category Pure mathematics R&D Projects GJ19-06792Y GA ČR - Czech Science Foundation (CSF) Method of publishing Limited access Institutional support UIVT-O - RVO:67985807 UT WOS 000532684200006 EID SCOPUS 85081686829 DOI 10.1016/j.comgeo.2020.101629 Annotation We study Hamiltonicity for some of the most general variants of Delaunay and Gabriel graphs. Instead of defining these proximity graphs using circles, we use an arbitrary convex shape C. Let S be a point set in the plane. The k-order Delaunay graph of S, denoted k-DGC(S), has vertex set S, and edges defined as follows. Given p,q∈S, pq is an edge of k-DGC(S) provided there exists some homothet of C with p and q on its boundary and containing at most k points of S different from p and q. The k-order Gabriel graph, denoted k-GGC(S), is defined analogously, except that the homothets considered are restricted to be smallest homothets of C with p and q on the boundary. We provide upper bounds on the minimum value of k for which k-GGC(S) is Hamiltonian. Since k-GGC(S) ⊆ k-DGC(S), all results carry over to k-DGC(S). In particular, we give upper bounds of 24 for every C and 15 for every point-symmetric C. We also improve these bounds to 7 for squares, 11 for regular hexagons, 12 for regular octagons, and 11 for even-sided regular t-gons (for t≥10). These constitute the first general results on Hamiltonicity for convex shape Delaunay and Gabriel graphs. In addition, we show lower bounds of k=3 and k=6 on the existence of a bottleneck Hamiltonian cycle in the k-order Gabriel graph for squares and hexagons, respectively. Finally, we construct a point set such that for an infinite family of regular polygons Pt, the Delaunay graph DGPt does not contain a Hamiltonian cycle. Workplace Institute of Computer Science Contact Tereza Šírová, sirova@cs.cas.cz, Tel.: 266 053 800 Year of Publishing 2021 Electronic address http://dx.doi.org/10.1016/j.comgeo.2020.101629
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