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# Hamiltonicity for Convex Shape Delaunay and Gabriel Graphs

- 1.0523630 - ÚI 2021 RIV NL eng J - Journal Article
**Bose, P. - Cano, P. - Saumell, Maria - Silveira, R.I.**

Hamiltonicity for Convex Shape Delaunay and Gabriel Graphs.*Computational Geometry-Theory and Applications*. Roč. 89, August 2020 (2020), č. článku 101629. ISSN 0925-7721. E-ISSN 1879-081X**R&D Projects**: GA ČR(CZ) GJ19-06792Y**Institutional support**: RVO:67985807**Keywords**: Delaunay graphs * Hamiltonicity * Gabriel graphs**OECD category**: Pure mathematics**Impact factor**: 0.537, year: 2020**Method of publishing**: Limited access

http://dx.doi.org/10.1016/j.comgeo.2020.101629

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.

**Permanent Link:**http://hdl.handle.net/11104/0307945

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