Circumventing Connectivity for Kernelization

Classical vertex subset problems demanding connectivity are of the following form: given an input graph G on n vertices and an integer k, find a set S of at most k vertices that satisfies a property and G[S] is connected. In this paper, we initiate a systematic study of such problems under a specific connectivity constraint, from the viewpoint of Kernelization (Parameterized) Complexity. The specific form that we study does not demand that G[S] is connected but rather G[S] has a closed walk containing all the vertices in S. In particular, we study Closed Walk-Subgraph Vertex Cover (CW-SVC, in short), where given a graph G, a set \(X \subseteq E(G)\), and an integer k; the goal is to find a set of vertices S that hits all the edges in X and can be traversed by a closed walk of length at most k in G. When X is E(G), this corresponds to Closed Walk-Vertex Cover (CW-VC, in short). One can similarly define these variants for Feedback Vertex Set, namely Closed Walk-Subgraph Feedback Vertex Set (CW-SFVS, in short) and Closed Walk-Feedback Vertex Set (CW-FVS, in short). Our results are as follows: CW-VC and CW-SVC both admit a polynomial kernel, in contrast to Connected Vertex Cover that does not admit a polynomial kernel unless \(\mathsf{NP} \subseteq \mathsf{coNP}/\mathsf{poly}\). CW-FVS admits a polynomial kernel. On the other hand CW-SFVS does not admit even a polynomial Turing kernel unless the polynomial-time hierarchy collapses.