We study the kernelization complexity of structural parameterizations of the Vertex Cover problem. Here, the goal is to find a polynomial-time preprocessing algorithm that can reduce any instance (G,k) of the Vertex Cover problem to an equivalent one, whose size is polynomial in the size of a pre-determined complexity parameter of G. A long line of previous research deals with parameterizations based on the number of vertex deletions needed to reduce G to a member of a simple graph class ℱ, such as forests, graphs of bounded tree-depth, and graphs of maximum degree two. We set out to find the most general graph classes ℱ for which Vertex Cover parameterized by the vertex-deletion distance of the input graph to ℱ, admits a polynomial kernelization. We give a complete characterization of the minor-closed graph families ℱ for which such a kernelization exists. We introduce a new graph parameter called bridge-depth, and prove that a polynomial kernelization exists if and only if ℱ has bounded bridge-depth. The proof is based on an interesting connection between bridge-depth and the size of minimal blocking sets in graphs, which are vertex sets whose removal decreases the independence number.
@InProceedings{bougeret_et_al:LIPIcs.ICALP.2020.16, author = {Bougeret, Marin and Jansen, Bart M. P. and Sau, Ignasi}, title = {{Bridge-Depth Characterizes Which Structural Parameterizations of Vertex Cover Admit a Polynomial Kernel}}, booktitle = {47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)}, pages = {16:1--16:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-138-2}, ISSN = {1868-8969}, year = {2020}, volume = {168}, editor = {Czumaj, Artur and Dawar, Anuj and Merelli, Emanuela}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2020.16}, URN = {urn:nbn:de:0030-drops-124238}, doi = {10.4230/LIPIcs.ICALP.2020.16}, annote = {Keywords: vertex cover, parameterized complexity, polynomial kernel, structural parameterization, bridge-depth} }
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