## Abstract

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.

Original language | English |
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Article number | 112406 |

Journal | Discrete Mathematics |

Volume | 344 |

Issue number | 7 |

DOIs | |

State | Published - 1 Jul 2021 |

Externally published | Yes |

## Keywords

- Colored point set
- Enclosing simple polygon
- General bounds
- Particular cases

## ASJC Scopus subject areas

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics