TY - JOUR

T1 - Rainbow polygons for colored point sets in the plane

AU - Flores-Peñaloza, David

AU - Kano, Mikio

AU - Martínez-Sandoval, Leonardo

AU - Orden, David

AU - Tejel, Javier

AU - Tóth, Csaba D.

AU - Urrutia, Jorge

AU - Vogtenhuber, Birgit

N1 - Funding Information:
David Flores, David Orden, Javier Tejel, Jorge Urrutia, and Birgit Vogtenhuber are supported by the H2020-MSCA-RISE, EU project 734922-CONNECT . Research by David Flores-Peñaloza was supported by the grants UNAM PAPIIT, Mexico IN117317 and IN120520 . Research by Mikio Kano was supported by JSPS, Japan KAKENHI Grant Number 16K05248 . Research by Leonardo Martínez-Sandoval was supported by the grant ANR-17-CE40-0018 of the French National Research Agency ANR (project CAPPS). Research by David Orden was supported by project MTM2017-83750-P of the Spanish Ministry of Science (AEI/FEDER, UE) and by Project PID2019-104129GB-I00/AEI / 10.13039/501100011033 of the Spanish Ministry of Science and Innovation . Research by Javier Tejel was supported by MINECO, Spain project MTM2015-63791-R , Gobierno de Aragón, Spain under Grant E41-17R (FEDER), and Project PID2019-104129GB-I00/AEI / 10.13039/501100011033 of the Spanish Ministry of Science and Innovation . Research by Csaba D. Tóth was supported by NSF, United States awards CCF-1422311 , CCF-1423615 , and DMS-1800734 . Research by Jorge Urrutia was supported by UNAM, Mexico project PAPIIT IN102117 . Research by Birgit Vogtenhuber was supported by the Austrian Science Fund within the collaborative DACH project Arrangements and Drawings as FWF project I 3340-N35 .
Publisher Copyright:
© 2021 Elsevier B.V.

PY - 2021/7

Y1 - 2021/7

N2 - Given a colored point set in the plane, a perfect rainbow polygon is a simple polygon that contains exactly one point of each color, either in its interior or on its boundary. Let rb-index(S) denote the smallest size of a perfect rainbow polygon for a colored point set S, and let rb-index(k) be the maximum of rb-index(S) over all k-colored point sets in general position; that is, every k-colored point set S has a perfect rainbow polygon with at most rb-index(k) vertices. In this paper, we determine the values of rb-index(k) up to k=7, which is the first case where rb-index(k)≠k, and we prove that for k≥5, [Formula presented] Furthermore, for a k-colored set of n points in the plane in general position, a perfect rainbow polygon with at most [Formula presented] vertices can be computed in O(nlogn) time.

AB - Given a colored point set in the plane, a perfect rainbow polygon is a simple polygon that contains exactly one point of each color, either in its interior or on its boundary. Let rb-index(S) denote the smallest size of a perfect rainbow polygon for a colored point set S, and let rb-index(k) be the maximum of rb-index(S) over all k-colored point sets in general position; that is, every k-colored point set S has a perfect rainbow polygon with at most rb-index(k) vertices. In this paper, we determine the values of rb-index(k) up to k=7, which is the first case where rb-index(k)≠k, and we prove that for k≥5, [Formula presented] Furthermore, for a k-colored set of n points in the plane in general position, a perfect rainbow polygon with at most [Formula presented] vertices can be computed in O(nlogn) time.

KW - Colored point set

KW - Enclosing simple polygon

KW - General bounds

KW - Particular cases

UR - http://www.scopus.com/inward/record.url?scp=85104332397&partnerID=8YFLogxK

U2 - 10.1016/j.disc.2021.112406

DO - 10.1016/j.disc.2021.112406

M3 - Article

AN - SCOPUS:85104332397

VL - 344

JO - Discrete Mathematics

JF - Discrete Mathematics

SN - 0012-365X

IS - 7

M1 - 112406

ER -