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#### A Quartic Kernel for Pathwidth-One Vertex Deletion

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##### Citation

Philip, G., Raman, V., & Villanger, Y. (2010). A Quartic Kernel for Pathwidth-One
Vertex Deletion. In D. M. Thilikos (*Graph Theoretic Concepts
in Computer Science* (pp. 196-207). Berlin: Springer. doi:10.1007/978-3-642-16926-7_19.

Cite as: http://hdl.handle.net/11858/00-001M-0000-0019-DC03-C

##### Abstract

The pathwidth of a graph is a measure of how path-like the graph is.
Given a graph G and an integer k, the problem of finding whether
there exist at most k vertices in G whose deletion results in
a graph of pathwidth at most one is \npc{}. We initiate the study
of the parameterized complexity of this problem, parameterized by
k. We show that the problem has a quartic vertex-kernel: We show
that, given an input instance (G=(V,E),k);|V|=n, we can construct,
in polynomial time, an instance (G',k') such that (i) (G,k)
is a YES instance if and only if (G',k') is a YES instance, (ii)
G' has \Oh(k^4}) vertices, and (iii) k'≤ k. We also give
a fixed parameter tractable (FPT) algorithm for the problem that runs
in \Oh(7^{kk⋅ n²) time.