{"@context":"https:\/\/schema.org\/","@type":"ScholarlyArticle","@id":"#article18048","name":"Improved Bounds for Covering Paths and Trees in the Plane","abstract":"A covering path for a planar point set is a path drawn in the plane with straight-line edges such that every point lies at a vertex or on an edge of the path. A covering tree is defined analogously. Let \u03c0(n) be the minimum number such that every set of n points in the plane can be covered by a noncrossing path with at most \u03c0(n) edges. Let \u03c4(n) be the analogous number for noncrossing covering trees. Dumitrescu, Gerbner, Keszegh, and T\u00f3th (Discrete & Computational Geometry, 2014) established the following inequalities: 5n\/9 - O(1) < \u03c0(n) < (1-1\/601080391)n, and 9n\/17 - O(1) < \u03c4(n) \u2a7d \u230a5n\/6\u230b. We report the following improved upper bounds: \u03c0(n) \u2a7d (1-1\/22)n, and \u03c4(n) \u2a7d \u23084n\/5\u2309.\r\nIn the same context we study rainbow polygons. For a set of colored points in the plane, a perfect rainbow polygon is a simple polygon that contains exactly one point of each color in its interior or on its boundary. Let \u03c1(k) be the minimum number such that every k-colored point set in the plane admits a perfect rainbow polygon of size \u03c1(k). Flores-Pe\u00f1aloza, Kano, Mart\u00ednez-Sandoval, Orden, Tejel, T\u00f3th, Urrutia, and Vogtenhuber (Discrete Mathematics, 2021) proved that 20k\/19 - O(1) < \u03c1(k) < 10k\/7 + O(1). We report the improved upper bound of \u03c1(k) < 7k\/5 + O(1). \r\nTo obtain the improved bounds we present simple O(nlog n)-time algorithms that achieve paths, trees, and polygons with our desired number of edges.","keywords":["planar point sets","covering paths","covering trees","rainbow polygons"],"author":{"@type":"Person","name":"Biniaz, Ahmad","givenName":"Ahmad","familyName":"Biniaz","email":"mailto:abiniaz@uwindsor.ca","affiliation":"School of Computer Science, University of Windsor, Canada"},"position":19,"pageStart":"19:1","pageEnd":"19:15","dateCreated":"2023-06-09","datePublished":"2023-06-09","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/legalcode","copyrightHolder":{"@type":"Person","name":"Biniaz, Ahmad","givenName":"Ahmad","familyName":"Biniaz","email":"mailto:abiniaz@uwindsor.ca","affiliation":"School of Computer Science, University of Windsor, Canada"},"copyrightYear":"2023","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/LIPIcs.SoCG.2023.19","funding":"Research supported by NSERC.","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":{"@type":"PublicationVolume","@id":"#volume18027","volumeNumber":258,"name":"39th International Symposium on Computational Geometry (SoCG 2023)","dateCreated":"2023-06-09","datePublished":"2023-06-09","editor":[{"@type":"Person","name":"Chambers, Erin W.","givenName":"Erin W.","familyName":"Chambers","email":"mailto:erin.chambers@slu.edu","sameAs":"https:\/\/orcid.org\/0000-0001-8333-3676","affiliation":"Saint Louis University, USA"},{"@type":"Person","name":"Gudmundsson, Joachim","givenName":"Joachim","familyName":"Gudmundsson","email":"mailto:joachim.gudmundsson@sydney.edu.au","sameAs":"https:\/\/orcid.org\/0000-0002-6778-7990","affiliation":"University of Sydney, Australia"}],"isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#article18048","isPartOf":{"@type":"Periodical","@id":"#series116","name":"Leibniz International Proceedings in Informatics","issn":"1868-8969","isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#volume18027"}}}