In this paper, we study the following problem: given a connected graph G, can we reduce the domination number of G by at least one using k edge contractions, for some fixed integer k >= 0? We show that for k <= 2, the problem is coNP-hard. We further prove that for k=1, the problem is W[1]-hard parameterized by the size of a minimum dominating set plus the mim-width of the input graph, and that it remains NP-hard when restricted to P_9-free graphs, bipartite graphs and {C_3,...,C_{l}}-free graphs for any l >= 3. Finally, we show that for any k >= 1, the problem is polynomial-time solvable for P_5-free graphs and that it can be solved in FPT-time and XP-time when parameterized by tree-width and mim-width, respectively.
@InProceedings{galby_et_al:LIPIcs.MFCS.2019.41, author = {Galby, Esther and Lima, Paloma T. and Ries, Bernard}, title = {{Reducing the Domination Number of Graphs via Edge Contractions}}, booktitle = {44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)}, pages = {41:1--41:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-117-7}, ISSN = {1868-8969}, year = {2019}, volume = {138}, editor = {Rossmanith, Peter and Heggernes, Pinar and Katoen, Joost-Pieter}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2019.41}, URN = {urn:nbn:de:0030-drops-109856}, doi = {10.4230/LIPIcs.MFCS.2019.41}, annote = {Keywords: domination number, blocker problem, graph classes} }
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