On the Small Cycle Transversal of Planar Graphs

We consider the problem of finding a k-edge transversal set that covers 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 linear kernel of size 36sk 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. Unlike the previous results on linear kernels of problems on planar graphs, our results are not subsumed by the recent meta-theorems on kernelization of Bodlaender et al. [FOCS, 2009 ].