Abstract
A dominating target of a graph G=(V,E) is a set of vertices T s.t. for all W ⊆ V, if T ⊆ W and induced subgraph on W is connected, then W is a dominating set of G. The size of the smallest dominating target is called dominating target number of the graph, dt(G). We provide polynomial time algorithms for minimum connected dominating set, Steiner set, and Steiner connected dominating set in dominating-pair graphs (i.e., dt(G)=2). We also give approximation algorithm for minimum connected dominating set with performance ratio 2 on graphs with small dominating targets. This is a significant improvement on appx ≤d(opt + 2) given by Fomin et.al. [2004] on graphs with small d-octopus.
Classification: Dominating target, d-octopus, Dominating set, Dominating-pair graph, Steiner tree.
Chapter PDF
Similar content being viewed by others
References
Guha, S., Khuller, S.: Approximation Algorithms for Connected Dominating Sets. In: Díaz, J. (ed.) ESA 1996. LNCS, vol. 1136, pp. 179–193. Springer, Heidelberg (1996)
Balakrishnan, H., Rajaraman, A., Rangan, C.P.: Connected Domination and Steiner Set on Asteroidal Triple-Free Graphs. In: Dehne, F., Sack, J.-R., Santoro, N. (eds.) WADS 1993. LNCS, vol. 709, pp. 131–141. Springer, Heidelberg (1993)
Robins, G., Zelikovsky, A.: Improved Steiner Tree Approximation in Graphs. In: Proceedings of SODA. LNCS, pp. 770–779. Springer, Heidelberg (2000)
Guha, S., Khuller, S.: Improved Methods for Approximating Node Weighted Steiner Trees and Connected Dominating Sets. In: Proceedings of FSTTCS year. LNCS, pp. 54–65. Springer, Heidelberg (1998)
Prömel, H.J., Steger, A.: RNC-Approximation Algorithms for the Steiner Problem. In: Proceedings of STACS. LNCS, pp. 559–570. Springer, Heidelberg (1997)
Clementi, A.E.F., Trevisan, L.: Improved Non-Approximability Results for Vertex Cover with Density Constraints. In: Cai, J.-Y., Wong, C.K. (eds.) COCOON 1996. LNCS, vol. 1090, pp. 333–342. Springer, Heidelberg (1996)
Garey, M.R., Johnson, D.S.: Computers and Intractability. Freeman, San Francisco (1978)
Ausiello, G., Crescenzi, P., Gambosi, G., Kann, V., Marchetti-Spaccamela, A., Protasi, M.: Complexity and Approximation. Springer, Heidelberg (2003)
Kahng, A.B., Robins, G.: On Optimal Interconnections for VLSI. Kluwer Academic, Dordrecht (1995)
Brandstädt, A., Lee, V.B., Spinrad, J.P.: Graph Classes: A Survey. SIAM Monographs on Discrete Mathematics and Applications (1999)
Caldwell, A., Kahng, A., Mantik, S., Markov, I., Zelikovsky, A.: On Wirelength Estimations for Row-Biased Placement. In: Proceedings of International Symposium on Physical Design, pp. 4–11 (1998)
Kloks, T., Kratsch, D., Müller, H.: On the Structure of Graphs with Bounded Asteroidal Number. Graphs and Combinatorics 17, 295–306 (2001)
Korte, B., Prömel, H.J., Steger, A.: Steiner Trees in VLSI Layouts. J. Paths flows and VLSI layout (1990)
Cheng, X., Huang, X., Li, D., Wu, W., Du, D.-Z.: A Polynomial-Time Approximation Scheme for the Minimum-Connected Dominating Set in Ad Hoc Wireless Networks. Journal of Networks 42(4), 202–208 (2003)
Corneil, D.G., Olariu, S., Stewart, L.: Asteroidal Triple-Free Graphs. SIAM J. Discrete Math. 10(3), 399–430 (1997)
Even, S., Pnueli, A., Lempel, A.: Permutation Graphs and Transitive Graphs. J. ACM 19(3), 400–410 (1972)
Lekkerkerker, C.G., Boland, J.Ch.: Boland: Representation of a Finite Graphs by a Set of Intervals on the Real Line. J. Fund. Math. 51, 245–264 (1962)
Motwani, R.: Lecture Notes on Approximation Algorithms, Dept. of Comp. Sc., Stanford University, vol. I (1992)
Fomin, F.V., Kratsch, D., Müller, H.: Algorithms for Graphs with Small Octopus. Journal of Discrete Applied Mathematics 134, 105–128 (2004)
Kratsch, D., Spinrad, J.: Between O(nm) and O(n α). In: Prodeedings of SODA. LNCS, pp. 709–716. Springer, Heidelberg (2003)
Habib, M.: Substitution des Structures Combinatoires, Theorie et Algorithmes. These D’etat, Paris VI (1981)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2006 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Aggarwal, D., Dubey, C.K., Mehta, S.K. (2006). Algorithms on Graphs with Small Dominating Targets. In: Asano, T. (eds) Algorithms and Computation. ISAAC 2006. Lecture Notes in Computer Science, vol 4288. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11940128_16
Download citation
DOI: https://doi.org/10.1007/11940128_16
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-49694-6
Online ISBN: 978-3-540-49696-0
eBook Packages: Computer ScienceComputer Science (R0)