Before revision:  We present polynomial-time kernelizations for Cluster Editing/Deletion , Path Contractions and Feedback Vertex Set that run with O(\mathrm{poly (k) \log n) bits and compute a kernel of size polynomial in k . By first executing the new kernelizations and subsequently the best known polynomial-time kernelizations for the problem under consideration, we obtain the best known kernels in polynomial time with O( \mathrm{poly(k) \log n) bits.
After revision:  We present a full kernel for Path Contraction and Cluster Editing/Deletion , Path Contractions and Feedback Vertex Set that run with O(\mathrm{poly (k) \log n) bits and compute a kernel of size polynomial in k . By first executing the new kernelizations and subsequently the best known polynomial-time kernelizations for the problem under consideration, we obtain the best known kernels in polynomial time with O( \mathrm{poly(k) \log n) bits.