Abstract
Given two sets σ, ρ of nonnegative integers, a set S of vertices of a graph G is (σ,ρ)-dominating if |S ∩ N(v)| ∈ σ for every vertex v ∈ S, and |S ∩ N(v)| ∈ ρ for every v ∉ S. This concept, introduced by Telle in 1990’s, generalizes and unifies several variants of graph domination studied separately before. We study the parameterized complexity of (σ,ρ)-domination in this general setting. Among other results we show that existence of a (σ,ρ)-dominating set of size k (and at most k) are W[1]-complete problems (when parameterized by k) for any pair of finite sets σ and ρ. We further present results on dual parametrization by n − k, and results on certain infinite sets (in particular for σ, ρ being the sets of even and odd integers).
Access provided by Autonomous University of Puebla. Download to read the full chapter text
Chapter PDF
Similar content being viewed by others
References
Alon, N., Gutner, S.: Linear time algorithms for finding a dominating set of fixed size in degenerated graphs. In: Lin, G. (ed.) COCOON 2007. LNCS, vol. 4598, pp. 394–405. Springer, Heidelberg (2007)
Bodlaender, H.L., Kratsch, D.: A note on fixed parameter intractability of some domination-related problems. Private communication (1994)
Cesati, M.: Perfect code is W[1]-complete. Inf. Process. Lett. 81, 163–168 (2002)
Dawar, A., Grohe, M., Kreutzer, S.: Locally excluding a minor. In: LICS 2007, pp. 270–279. IEEE Computer Society Press, Los Alamitos (2007)
Downey, R.G., Fellows, M.R.: Fixed-parameter tractability and completeness. Congressus Numerantium, 161–178 (1992)
Downey, R.G., Fellows, M.R.: Fixed-parameter tractability and completeness. II. On completeness for W[1]. Theoret. Comput. Sci. 141, 109–131 (1995)
Downey, R.G., Fellows, M.R.: Parameterized Complexity. Springer, Heidelberg (1998)
Downey, R.G., Fellows, M.R.: Threshold dominating sets and an improved characterization of W[2]. Theoret. Comput. Sci. 209, 123–140 (1998)
Downey, R.G., Fellows, M.R., Vardy, A., Whittle, G.: The parametrized complexity of some fundamental problems in coding theory. SIAM J. Comput. 29 (electronic), 545–570 (1999)
Flum, J., Grohe, M.: Parameterized Complexity Theory. Springer, Heidelberg (2006)
Golovach, P.A., Kratochvíl, J.: Computational complexity of generalized domination: A complete dichotomy for chordal graphs. In: Brandstädt, A., Kratsch, D., Müller, H. (eds.) WG 2007. LNCS, vol. 4769, pp. 1–11. Springer, Heidelberg (2007)
Golovach, P.A., Kratochvíl, J.: Generalized domination in degenerate graphs: a complete dichotomy of computational complexity. In: Agrawal, M., Du, D.-Z., Duan, Z., Li, A. (eds.) TAMC 2008. LNCS, vol. 4978, pp. 182–191. Springer, Heidelberg (2008)
Heggernes, P., Telle, J.A.: Partitioning graphs into generalized dominating sets. Nordic J. Comput. 5, 128–142 (1998)
Kloks, T., Cai, L.: Parameterized tractability of some (efficient) Y - domination variants for planar graphs and t-degenerate graphs. In: Proceedings of the International Computer Symposium (ICS 2000), Taiwan (2000)
Kratochvíl, J., Manuel, P.D., Miller, M.: Generalized domination in chordal graphs. Nordic J. Comput. 2, 41–50 (1995)
Moser, H., Thilikos, D.M.: Parameterized complexity of finding regular induced subgraphs. In: Broersma, H., Dantchev, S.S., Johnson, M., Szeider, S. (eds.) ACiD 2006. Texts in Algorithmics, vol. 7, pp. 107–118. King’s College, London (2006)
Niedermeier, R.: Invitation to Fixed Parameter Algorithms. Oxford University Press, USA (2006)
Telle, J.A.: Complexity of domination-type problems in graphs. Nordic J. Comput. 1, 157–171 (1994)
Telle, J.A.: Vertex partitioning problems: characterization, complexity and algorithms on partial k-trees. PhD thesis. Department of Computer Science, University of Oregon, Eugene (1994)
Telle, J.A., Proskurowski, A.: Algorithms for vertex partitioning problems on partial k-trees. SIAM J. Discrete Math. 10, 529–550 (1997)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2010 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Golovach, P.A., Kratochvíl, J., Suchý, O. (2010). Parameterized Complexity of Generalized Domination Problems. In: Paul, C., Habib, M. (eds) Graph-Theoretic Concepts in Computer Science. WG 2009. Lecture Notes in Computer Science, vol 5911. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-11409-0_12
Download citation
DOI: https://doi.org/10.1007/978-3-642-11409-0_12
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-11408-3
Online ISBN: 978-3-642-11409-0
eBook Packages: Computer ScienceComputer Science (R0)