{"@context":"https:\/\/schema.org\/","@type":"ScholarlyArticle","@id":"#article8194","name":"An FPT Algorithm and a Polynomial Kernel for Linear Rankwidth-1 Vertex Deletion","abstract":"Linear rankwidth is a linearized variant of rankwidth, introduced by Oum and Seymour [Approximating clique-width and branch-width. J. Combin. Theory Ser. B, 96(4):514-528, 2006.], and it is similar to pathwidth, which is the linearized variant of treewidth. Motivated from the results on graph modification problems into graphs of bounded treewidth or pathwidth, we investigate a graph modification problem into the class of graphs having linear rankwidth at most one, called the Linear Rankwidth-1 Vertex Deletion (shortly, LRW1-Vertex Deletion). In this problem, given an n-vertex graph G and a positive integer k, we want to decide whether there is a set of at most k vertices whose removal turns G into a graph of linear rankwidth at most one and if one exists, find such a vertex set. While the meta-theorem of Courcelle, Makowsky, and Rotics implies thatLRW1-Vertex Deletion can be solved in time f(k) * n^3 for some function f, it is not clear whether this problem allows a runtime with a modest exponential function. We establish that LRW1-Vertex Deletion can be solved in time 8^k * n^{O(1)}. The major obstacle to this end is how to handle a long induced cycle as an obstruction. To fix this issue, we define the necklace graphs and investigate their structural properties.\r\nWe also show that the LRW1-Vertex Deletion has a polynomial kernel.","keywords":["(linear) rankwidth","distance-hereditary graphs","thread graphs","parameterized complexity","kernelization"],"author":[{"@type":"Person","name":"Kant\u00e9, Mamadou Moustapha","givenName":"Mamadou Moustapha","familyName":"Kant\u00e9"},{"@type":"Person","name":"Kim, Eun Jung","givenName":"Eun Jung","familyName":"Kim"},{"@type":"Person","name":"Kwon, O-joung","givenName":"O-joung","familyName":"Kwon"},{"@type":"Person","name":"Paul, Christophe","givenName":"Christophe","familyName":"Paul"}],"position":12,"pageStart":138,"pageEnd":150,"dateCreated":"2015-11-19","datePublished":"2015-11-19","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Kant\u00e9, Mamadou Moustapha","givenName":"Mamadou Moustapha","familyName":"Kant\u00e9"},{"@type":"Person","name":"Kim, Eun Jung","givenName":"Eun Jung","familyName":"Kim"},{"@type":"Person","name":"Kwon, O-joung","givenName":"O-joung","familyName":"Kwon"},{"@type":"Person","name":"Paul, Christophe","givenName":"Christophe","familyName":"Paul"}],"copyrightYear":"2015","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/LIPIcs.IPEC.2015.138","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":{"@type":"PublicationVolume","@id":"#volume6246","volumeNumber":43,"name":"10th International Symposium on Parameterized and Exact Computation (IPEC 2015)","dateCreated":"2015-11-19","datePublished":"2015-11-19","editor":[{"@type":"Person","name":"Husfeldt, Thore","givenName":"Thore","familyName":"Husfeldt"},{"@type":"Person","name":"Kanj, Iyad","givenName":"Iyad","familyName":"Kanj"}],"isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#article8194","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":"#volume6246"}}}