In this paper we study the kernelization of the d-Path Vertex Cover (d-PVC) problem. Given a graph G, the problem requires finding whether there exists a set of at most k vertices whose removal from G results in a graph that does not contain a path (not necessarily induced) with d vertices. It is known that d-PVC is NP-complete for d ≥ 2. Since the problem generalizes to d-Hitting Set, it is known to admit a kernel with 𝒪(dk^d) edges. We improve on this by giving better kernels. Specifically, we give kernels with 𝒪(k²) vertices and edges for the cases when d = 4 and d = 5. Further, we give a kernel with 𝒪(k⁴d^{2d+9}) vertices and edges for general d.
@InProceedings{cerveny_et_al:LIPIcs.MFCS.2022.29, author = {\v{C}erven\'{y}, Radovan and Choudhary, Pratibha and Such\'{y}, Ond\v{r}ej}, title = {{On Kernels for d-Path Vertex Cover}}, booktitle = {47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)}, pages = {29:1--29:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-256-3}, ISSN = {1868-8969}, year = {2022}, volume = {241}, editor = {Szeider, Stefan and Ganian, Robert and Silva, Alexandra}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2022.29}, URN = {urn:nbn:de:0030-drops-168279}, doi = {10.4230/LIPIcs.MFCS.2022.29}, annote = {Keywords: Parameterized complexity, Kernelization, d-Hitting Set, d-Path Vertex Cover, Expansion Lemma} }
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