Kernels in planar digraphs

Authors:
Gregory Gutin;Ton Kloks;Chuan Min Lee;Anders Yeo
Affiliations:
Department of Computer Science, Royal Holloway, University of London, Egham, Surrey, TW20 0EX, UK;Department of Mathematics and Computer Science, The University of Lethbridge, Alb., Canada T1K 3M4;Department of Computer Science and Information Engineering, Chung Cheng University, Ming-Shiun, Chaiyi 621, Taiwan, ROC;Department of Computer Science, Royal Holloway, University of London, Egham, Surrey, TW20 0EX, UK
Venue:
Journal of Computer and System Sciences
Year:
2005

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Abstract

A set S of vertices in a digraph D=(V,A) is a kernel if S is independent and every vertex in V-S has an out-neighbor in S. We show that there exist O(n2^1^9^.^1^k+n^4)-time and O(k^3^6+2^1^9^.^1^kk^9+n^2)-time algorithms for checking whether a planar digraph D of order n has a kernel with at most k vertices. Moreover, if D has a kernel of size at most k, the algorithms find such a kernel of minimal size.