Polynomial kernels for paw-free edge modification problems

Given a graph G and an integer k, the paw-free completion problem asks whether it is possible to add at most k edges make paw-free. The edge deletion defined analogously. Sandeep Sivadasan (IPEC 2015) asked these problems admit polynomial kernels. We answer both questions affirmatively by presenting, respectively, O(k)-vertex O(k4)-vertex kernels for them. This part of ongoing program that aims understanding compressibility H-free modification general H.