Parameterized Algorithms for Generalized Domination

  • Authors:

  • Venkatesh Raman;Saket Saurabh;Sriganesh Srihari

  • Affiliations:

  • The Institute of Mathematical Sciences, , Chennai 600 113;Department of Informatics, University of Bergen, Bergen, Norway;School of Computing, National University of Singapore1, Singapore 117590

  • Venue:
  • COCOA 2008 Proceedings of the 2nd international conference on Combinatorial Optimization and Applications
  • Year:
  • 2008

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Abstract

We study the parameterized complexity of a generalization of Dominating Setproblem, namely, the Vector Dominating Setproblem. Here, given an undirected graph G= (V,E), with V= {v1, 驴 , vn}, a vector $\vec{l}=(l(v_1),\cdots, l(v_n))$ and an integer parameter k, the goal is to determine whether there exists a subset Dof at most kvertices such that for every vertex v驴 V驴 D, at least l(v) of its neighbors are in D. This problem encompasses the well studied problems --- Vertex Cover(when l(v) = d(v) for all v驴 V, where d(v) is the degree of vertex v) and Dominating Set(when l(v) = 1 for all v驴 V). While Vertex Coveris known to be fixed parameter tractable, Dominating Setis known to be W[2]-complete. In this paper, we identify vectors based on several measures for which this generalized problem is fixed parameter tractable and W-hard. We also show that the Vector Dominating Setis fixed parameter tractable for graphs of bounded degeneracy and for graphs excluding cycles of length four.