Hamiltonicity for Convex Shape Delaunay and Gabriel Graphs
The result's identifiers
Result code in IS VaVaI
<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F67985807%3A_____%2F20%3A00523630" target="_blank" >RIV/67985807:_____/20:00523630 - isvavai.cz</a>
Alternative codes found
RIV/68407700:21240/20:00347911
Result on the web
<a href="http://dx.doi.org/10.1016/j.comgeo.2020.101629" target="_blank" >http://dx.doi.org/10.1016/j.comgeo.2020.101629</a>
DOI - Digital Object Identifier
<a href="http://dx.doi.org/10.1016/j.comgeo.2020.101629" target="_blank" >10.1016/j.comgeo.2020.101629</a>
Alternative languages
Result language
angličtina
Original language name
Hamiltonicity for Convex Shape Delaunay and Gabriel Graphs
Original language description
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.
Czech name
—
Czech description
—
Classification
Type
J<sub>imp</sub> - Article in a specialist periodical, which is included in the Web of Science database
CEP classification
—
OECD FORD branch
10101 - Pure mathematics
Result continuities
Project
<a href="/en/project/GJ19-06792Y" target="_blank" >GJ19-06792Y: Structural properties of visibility in terrains and farthest color Voronoi diagrams</a><br>
Continuities
I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace
Others
Publication year
2020
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů
Data specific for result type
Name of the periodical
Computational Geometry-Theory and Applications
ISSN
0925-7721
e-ISSN
—
Volume of the periodical
89
Issue of the periodical within the volume
August 2020
Country of publishing house
NL - THE KINGDOM OF THE NETHERLANDS
Number of pages
17
Pages from-to
101629
UT code for WoS article
000532684200006
EID of the result in the Scopus database
2-s2.0-85081686829