{"@context":"https:\/\/schema.org\/","@type":"ScholarlyArticle","@id":"#article7893","name":"On-line Coloring between Two Lines","abstract":"We study on-line colorings of certain graphs given as intersection graphs of objects \"between two lines\", i.e., there is a pair of horizontal lines such that each object of the representation is a connected set contained in the strip between the lines and touches both. Some of the graph classes admitting such a representation are permutation graphs (segments), interval graphs (axis-aligned rectangles), trapezoid graphs (trapezoids) and cocomparability graphs (simple curves). We present an on-line algorithm coloring graphs given by convex sets between two lines that uses O(w^3) colors on graphs with maximum clique size w.\r\n\r\nIn contrast intersection graphs of segments attached to a single line may force any on-line coloring algorithm to use an arbitrary number of colors even when w=2.\r\n\r\nThe left-of relation makes the complement of intersection graphs of objects between two lines into a poset. As an aside we discuss the relation of the class C of posets obtained from convex sets between two lines with some other classes of posets: all 2-dimensional posets and all posets of height 2 are in C but there is a 3-dimensional poset of height 3 that does not belong to C.\r\n\r\nWe also show that the on-line coloring problem for curves between two lines is as hard as the on-line chain partition problem for arbitrary posets.","keywords":["intersection graphs","cocomparability graphs","on-line coloring"],"author":[{"@type":"Person","name":"Felsner, Stefan","givenName":"Stefan","familyName":"Felsner"},{"@type":"Person","name":"Micek, Piotr","givenName":"Piotr","familyName":"Micek"},{"@type":"Person","name":"Ueckerdt, Torsten","givenName":"Torsten","familyName":"Ueckerdt"}],"position":48,"pageStart":630,"pageEnd":641,"dateCreated":"2015-06-12","datePublished":"2015-06-12","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Felsner, Stefan","givenName":"Stefan","familyName":"Felsner"},{"@type":"Person","name":"Micek, Piotr","givenName":"Piotr","familyName":"Micek"},{"@type":"Person","name":"Ueckerdt, Torsten","givenName":"Torsten","familyName":"Ueckerdt"}],"copyrightYear":"2015","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/LIPIcs.SOCG.2015.630","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":{"@type":"PublicationVolume","@id":"#volume6237","volumeNumber":34,"name":"31st International Symposium on Computational Geometry (SoCG 2015)","dateCreated":"2015-06-12","datePublished":"2015-06-12","editor":[{"@type":"Person","name":"Arge, Lars","givenName":"Lars","familyName":"Arge"},{"@type":"Person","name":"Pach, J\u00e1nos","givenName":"J\u00e1nos","familyName":"Pach"}],"isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#article7893","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":"#volume6237"}}}