Abstract
The Power Dominating Set problem is an extension of the well-known domination problem on graphs in a way that we enrich it by a second propagation rule: Given a graph G(V,E) a set P ⊆ V is a power dominating set if every vertex is observed after we have applied the next two rules exhaustively. First, a vertex is observed if v ∈ P or it has a neighbor in P. Secondly, if an observed vertex has exactly one unobserved neighbor u, then also u will be observed as well. We show that Power Dominating Set remains \(\mathcal{NP}\)-hard on cubic graphs. We designed an algorithm solving this problem in time \(\mathcal{O}^*(1.7548^n)\) on general graphs. To achieve this we have used a new notion of search trees called reference search trees providing non-local pointers.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Aazami, A., Stilp, M.D.: Approximation algorithms and hardness for domination with propagation. In: Charikar, M., Jansen, K., Reingold, O., Rolim, J.D.P. (eds.) RANDOM 2007 and APPROX 2007. LNCS, vol. 4627, pp. 1–15. Springer, Heidelberg (2007)
Fernau, H., Raible, D.: Exact algorithms for maximum acyclic subgraph on a superclass of cubic graphs. In: Nakano, S.-i., Rahman, M. S. (eds.) WALCOM 2008. LNCS, vol. 4921, pp. 144–156. Springer, Heidelberg (2008); long version available as Technical Report 08-5, Technical Reports Mathematics / Computer Science, University of Trier, Germany (2008)
Fomin, F.V., Grandoni, F., Kratsch, D.: Measure and conquer: domination – a case study. In: Caires, L., Italiano, G.F., Monteiro, L., Palamidessi, C., Yung, M. (eds.) ICALP 2005. LNCS, vol. 3580, pp. 191–203. Springer, Heidelberg (2005)
Fomin, F.V., Grandoni, F., Kratsch, D.: Solving connected dominating set faster than 2n. In: Arun-Kumar, S., Garg, N. (eds.) FSTTCS 2006. LNCS, vol. 4337, pp. 152–163. Springer, Heidelberg (2006)
Garey, M.R., Johnson, D.S.: The rectilinear Steiner tree problem is NP-complete. SIAM J. Appl. Math. 32(4), 826–834 (1976)
Guo, J., Niedermeier, R., Raible, D.: Improved algorithms and complexity results for power domination in graphs. In: Liśkiewicz, M., Reischuk, R. (eds.) FCT 2005. LNCS, vol. 3623, pp. 172–184. Springer, Heidelberg (2005)
Haynes, T.W., Mitchell Hedetniemi, S., Hedetniemi, S.T., Henning, M.A.: Domination in graphs applied to electric power networks. SIAM J. Discrete Math. 15(4), 519–529 (2002)
Kneis, J., Mölle, D., Richter, S., Rossmanith, P.: Parameterized power domination complexity. Inf. Process. Lett. 98(4), 145–149 (2006)
Liao, C.-S., Lee, D.-T.: Power domination problem in graphs. In: Wang, L. (ed.) COCOON 2005. LNCS, vol. 3595, pp. 818–828. Springer, Heidelberg (2005)
Speckennmeyer, E.: On feedback vertex sets and nonseparating independent sets in cubic graphs. Journal of Graph Theory 3, 405–412 (1988)
van Rooij, J.M.M., Bodlaender, H.L.: Design by measure and conquer, a faster exact algorithm for dominating set. In: STACS 2008, volume 08001 of Dagstuhl Seminar Proceedings. Internationales Begegnungs- und Forschungszentrum fuer Informatik (IBFI), Schloss Dagstuhl, Germany, pp. 657–668 (2008)
Xu, G., Kang, L., Shan, E., Zhao, M.: Power domination in block graphs. Theor. Comput. Sci. 359(1-3), 299–305 (2006)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2008 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Raible, D., Fernau, H. (2008). Power Domination in \(\mathcal{O}^*(1.7548^n)\) Using Reference Search Trees. In: Hong, SH., Nagamochi, H., Fukunaga, T. (eds) Algorithms and Computation. ISAAC 2008. Lecture Notes in Computer Science, vol 5369. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-92182-0_15
Download citation
DOI: https://doi.org/10.1007/978-3-540-92182-0_15
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-92181-3
Online ISBN: 978-3-540-92182-0
eBook Packages: Computer ScienceComputer Science (R0)