Kernelization algorithm for the following problem We are given an undirected graph $ G $ and a positive parameter
$ k \geq 0 $. The problem is to decide if there exists a set $ S \subseteq V(G) $ of size at most $ k $ such that $ G − S $ does not have any path on three vertices.
Is there any deterministic kernel algorithm for this problem with $ \Theta (k^2) $ vertices?
Kernelization algorithm definition: Given $ (G,k) $, output an equivalent instance $ (G’,k’) $ of the same problem in polynomial time, such that $ |G’| \leq f(k) $ and $ k’ \leq k $.
Your problem can be stated as $3$-path vertex cover problem. A simple google search gives this paper. The authors design a kernel of size $5k$ for this problem.