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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 [Approxi-mating 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 that LRW1-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.
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Submitted on : Thursday, November 15, 2018 - 7:00:14 PM
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Christophe Paul, Eun Jung Kim, Mamadou Moustapha Kanté, O-Joung Kwon. An FPT Algorithm and a Polynomial Kernel for Linear Rankwidth-1 Vertex Deletion. IPEC: International symposium on Parameterized and Exact Computation, Sep 2015, Patras, Greece. pp.139-150, ⟨10.4230/LIPIcs.IPEC.2015.138⟩. ⟨lirmm-01264011⟩



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