Given a fixed graph H, the H-free editing problem asks whether we can edit at most k edges to make a graph contain no induced copy of H. We obtain a polynomial kernel for this problem when H is a diamond. The incompressibility dichotomy for H being a 3-connected graph and the classical complexity dichotomy suggest that except for H being a complete/empty graph, H-free editing problems admit polynomial kernels only for a few small graphs H. Therefore, we believe that our result is an essential step toward a complete dichotomy on the compressibility of H-free editing. Additionally, we give a cubic-vertex kernel for the diamond-free edge deletion problem, which is far simpler than the previous kernel of the same size for the problem.
@InProceedings{cao_et_al:LIPIcs.ESA.2018.10, author = {Cao, Yixin and Rai, Ashutosh and Sandeep, R. B. and Ye, Junjie}, title = {{A Polynomial Kernel for Diamond-Free Editing}}, booktitle = {26th Annual European Symposium on Algorithms (ESA 2018)}, pages = {10:1--10:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-081-1}, ISSN = {1868-8969}, year = {2018}, volume = {112}, editor = {Azar, Yossi and Bast, Hannah and Herman, Grzegorz}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2018.10}, URN = {urn:nbn:de:0030-drops-94732}, doi = {10.4230/LIPIcs.ESA.2018.10}, annote = {Keywords: Kernelization, Diamond-free, H-free editing, Graph modification problem} }
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