Parameterized Complexity of the Anchored kCore Problem for Directed Graphs
Abstract
Bhawalkar, Kleinberg, Lewi, Roughgarden, and Sharma [ICALP 2012] introduced the Anchored kCore problem, where the task is for a given 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 (called anchors) of H are of degree at least k. In this paper, we extend the notion of kcore to directed graphs and provide a number of new algorithmic and complexity results for the directed version of the problem. We show that  The decision version of the problem is NPcomplete 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.
 Publication:

arXiv eprints
 Pub Date:
 April 2013
 arXiv:
 arXiv:1304.5870
 Bibcode:
 2013arXiv1304.5870C
 Keywords:

 Computer Science  Data Structures and Algorithms;
 F.2.2;
 G.2.2