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An FPT Algorithm and a Polynomial Kernel for Linear Rankwidth-1 Vertex Deletion Kanté, Mamadou Moustapha Kim, Eun Jung Kwon, O-joung Paul, Christophe (linear) rankwidth distance-hereditary graphs thread graphs parameterized complexity kernelization 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. We also show that the LRW1-Vertex Deletion has a polynomial kernel. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Mamadou Moustapha Kanté and Eun Jung Kim and O-joung Kwon and Christophe Paul 2015 Is Part Of LIPIcs, Volume 43, 10th International Symposium on Parameterized and Exact Computation (IPEC 2015) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.IPEC.2015.138 urn:nbn:de:0030-drops-55788 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2015.138 eng https://creativecommons.org/licenses/by/3.0/legalcode