An FPT Algorithm and a Polynomial Kernel for Linear Rankwidth-1 Vertex Deletion
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.
(linear) rankwidth
distance-hereditary graphs
thread graphs
parameterized complexity
kernelization
138-150
Regular Paper
Mamadou Moustapha
KantÃ©
Mamadou Moustapha KantÃ©
Eun Jung
Kim
Eun Jung Kim
O-joung
Kwon
O-joung Kwon
Christophe
Paul
Christophe Paul
10.4230/LIPIcs.IPEC.2015.138
Creative Commons Attribution 3.0 Unported license
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