Hamiltonicity for Convex Shape Delaunay and Gabriel Graphs

Kód výsledku v 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>

Nalezeny alternativní kódy

RIV/68407700:21240/20:00347911

Výsledek na webu

<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>

Jazyk výsledku

angličtina

Název v původním jazyce

Hamiltonicity for Convex Shape Delaunay and Gabriel Graphs

Popis výsledku v původním jazyce

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.

Název v anglickém jazyce

Hamiltonicity for Convex Shape Delaunay and Gabriel Graphs

Popis výsledku anglicky

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.

Druh

J_imp - Článek v periodiku v databázi Web of Science

CEP obor

—

OECD FORD obor

10101 - Pure mathematics

Projekt

<a href="/cs/project/GJ19-06792Y" target="_blank" >GJ19-06792Y: Strukturální vlastnosti viditelnosti terénů a Voroného diagramů nejvzdálenější barvy</a><br>

Návaznosti

I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Rok uplatnění

2020

Kód důvěrnosti údajů

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Název periodika

Computational Geometry-Theory and Applications

ISSN

0925-7721

e-ISSN

—

Svazek periodika

89

Číslo periodika v rámci svazku

August 2020

Stát vydavatele periodika

NL - Nizozemsko

Počet stran výsledku

17

Strana od-do

101629

Kód UT WoS článku

000532684200006

EID výsledku v databázi Scopus

2-s2.0-85081686829