A diamond is a graph obtained by removing an edge from a complete graph on four vertices. A graph is diamond-free if it does not contain an induced diamond. The Diamond-free Edge Deletion problem asks to find whether there exist at most k edges in the input graph whose deletion results in a diamond-free graph. The problem was proved to be NP-complete and a polynomial kernel of O(k^4) vertices was found by Fellows et. al. (Discrete Optimization, 2011). In this paper, we give an improved kernel of O(k^3) vertices for Diamond-free Edge Deletion. We give an alternative proof of the NP-completeness of the problem and observe that it cannot be solved in time 2^{o(k)} * n^{O(1)}, unless the Exponential Time Hypothesis fails.
@InProceedings{sandeep_et_al:LIPIcs.IPEC.2015.365, author = {Sandeep, R. B. and Sivadasan, Naveen}, title = {{Parameterized Lower Bound and Improved Kernel for Diamond-free Edge Deletion}}, booktitle = {10th International Symposium on Parameterized and Exact Computation (IPEC 2015)}, pages = {365--376}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-92-7}, ISSN = {1868-8969}, year = {2015}, volume = {43}, editor = {Husfeldt, Thore and Kanj, Iyad}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2015.365}, URN = {urn:nbn:de:0030-drops-55976}, doi = {10.4230/LIPIcs.IPEC.2015.365}, annote = {Keywords: edge deletion problems, polynomial kernelization} }
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