Kernelization of the 33-path vertex cover problem

The 33-path vertex cover problem is an extension of the well-known vertex cover problem. It asks for a vertex set S⊆V(G)S⊆V(G) of minimum cardinality such that G−SG−S only contains independent vertices and edges. In this paper we will present a polynomial algorithm which computes two disjoint sets T1,T2T1,T2 of vertices of GG such that (i) for any 33-path vertex cover S′S′ in G[T2]G[T2], S′∪T1S′∪T1 is a 33-path vertex cover in GG, (ii) there exists a minimum 33-path vertex cover in GG which contains T1T1 and (iii) |T2|≤6⋅ψ3(G[T2])|T2|≤6⋅ψ3(G[T2]), where ψ3(G)ψ3(G) is the cardinality of a minimum 33-path vertex cover and T2T2 is the kernel of GG.