We consider the Directed Anchored k-Core problem, where the task is for a given directed graph G and integers b, k and p, to find an induced subgraph H with at least p vertices (the core) such that all but at most b vertices (the anchors) of H have in-degree at least k. For undirected graphs, this problem was introduced by Bhawalkar, Kleinberg, Lewi, Roughgarden, and Sharma [ICALP 2012]. We undertake a systematic analysis of the computational complexity of Directed Anchored k-Core and show that: - The decision version of the problem is NP-complete for every k>=1 even if the input graph is restricted to be a planar directed acyclic graph of maximum degree at most k+2. - The problem is fixed parameter tractable (FPT) parameterized by the size of the core p for k=1, and W[1]-hard for k>=2. - When the maximum degree of the graph is at most Delta, the problem is FPT parameterized by p+Delta if k>=Delta/2.
@InProceedings{chitnis_et_al:LIPIcs.FSTTCS.2013.79, author = {Chitnis, Rajesh and Fomin, Fedor V. and Golovach, Petr A.}, title = {{Parameterized Complexity of the Anchored k-Core Problem for Directed Graphs}}, booktitle = {IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2013)}, pages = {79--90}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-64-4}, ISSN = {1868-8969}, year = {2013}, volume = {24}, editor = {Seth, Anil and Vishnoi, Nisheeth K.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2013.79}, URN = {urn:nbn:de:0030-drops-43636}, doi = {10.4230/LIPIcs.FSTTCS.2013.79}, annote = {Keywords: Parameterized complexity, directed graphs, anchored \$k\$-core} }
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