On the small cycle transversal of planar graphs

Authors:
Ge Xia;Yong Zhang
Affiliations:
Department of Computer Science, Lafayette College, Easton, PA;Department of Computer Science, Kutztown University, Kutztown, PA
Venue:
WG'10 Proceedings of the 36th international conference on Graph-theoretic concepts in computer science
Year:
2010

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Abstract

We consider the problem of finding a k-edge transversal set that intersects all (simple) cycles of length at most s in a planar graph, where s ≥ 3 is a constant. This problem, referred to as SMALL CYCLE TRANSVERSAL, is known to be NP-complete. We present a polynomial-time algorithm that computes a kernel of size 36s3k for SMALL CYCLE TRANSVERSAL. In order to achieve this kernel, we extend the region decomposition technique of Alber et al. [J. ACM, 2004] by considering a unique region decomposition that is defined by shortest paths. Our kernel size is an exponential improvement in terms of s over the kernel size obtained under the meta-kernelization framework by Bodlaender et al. [FOCS, 2009].