, Volume 65, Issue 2, pp 443–466 | Cite as

Power Domination in Circular-Arc Graphs

  • Chung-Shou Liao
  • D. T. LeeEmail author


A set SV is a power dominating set (PDS) of a graph G=(V,E) if every vertex and every edge in G can be observed based on the observation rules of power system monitoring. The power domination problem involves minimizing the cardinality of a PDS of a graph. We consider this combinatorial optimization problem and present a linear time algorithm for finding the minimum PDS of an interval graph if the interval ordering of the graph is provided. In addition, we show that the algorithm, which runs in Θ(nlogn) time, where n is the number of intervals, is asymptotically optimal if the interval ordering is not given. We also show that the results hold for the class of circular-arc graphs.


Domination Power domination Interval graphs Circular-arc graphs Algorithm 



We wish to thank Hengchin Yeh, Ching-Chi Lin, G.J. Chang, and the anonymous reviewers for many valuable comments and suggestions, which helped us improve the quality of the presentation of the paper.


  1. 1.
    Aazami, A.: Domination in graphs with bounded propagation: algorithms formulations and hardness results. J. Comb. Optim. 19(4), 429–456 (2010) MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Aazami, A., Stilp, M.D.: Approximation algorithms and hardness for domination with propagation. SIAM J. Discrete Math. 23(3), 1382–1399 (2009) MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Atkins, D., Haynes, T.W., Henning, M.A.: Placing monitoring devices in electric power networks modelled by block graphs. Ars Comb. 79, 129–143 (2006) MathSciNetzbMATHGoogle Scholar
  4. 4.
    Baldwin, T.L., Mili, L., Boisen, M.B. Jr., Adapa, R.: Power system observability with minimal phasor measurement placement. IEEE Trans. Power Syst. 8(2), 707–715 (1993) CrossRefGoogle Scholar
  5. 5.
    Barrera, R.: On the power domination problem in graphs. M.S. Thesis, Texas State University, San Marcos, USA (2009) Google Scholar
  6. 6.
    Ben-Or, M.: Lower bounds for algebraic computation trees. In: Proc. 15th Annual Symposium on Theory of Computing, pp. 80–86 (1983) Google Scholar
  7. 7.
    Brueni, D.J.: Minimal PMU placement for graph observability: a decomposition approach. M.S. Thesis, Virginia Polytechnic Institute and State University, Blacksburg, USA (1993) Google Scholar
  8. 8.
    Brueni, D.J., Heath, L.S.: The PMU placement problem. SIAM J. Discrete Math. 19(3), 744–761 (2005) MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Chang, G.J.: Algorithmic aspects of domination in graphs. In: Du D.-Z., Pardalos P.M. (Eds.) Handbook of Combinatorial Optimization, vol. 3, pp. 339–405 (1998) Google Scholar
  10. 10.
    Dorbec, P., Mollard, M., Klavžar, S., Špacapan, S.: Power domination in product graphs. SIAM J. Discrete Math. 22(2), 554–567 (2008) MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Dorfling, M., Henning, M.A.: A note on power domination in grid graphs. Discrete Appl. Math. 154(6), 1023–1027 (2006) MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Emami, R., Abur, A.: Robust measurement design by placing synchronized phasor measurements on network branches. IEEE Trans. Power Syst. 25(1), 38–43 (2010) CrossRefGoogle Scholar
  13. 13.
    Gabow, H.N., Tarjan, R.E.: A linear-time algorithm for a special case of disjoint set union. J. Comput. Syst. Sci. 30(2), 209–221 (1985) MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Golumbic, M.C.: Algorithmic Graph Theory and Perfect Graphs. Academic Press, New York (1980) zbMATHGoogle Scholar
  15. 15.
    Guo, J., Niedermeier, R., Raible, D.: Improved algorithms and complexity results for power domination in graphs. Algorithmica 52(2), 177–202 (2008) MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Haynes, T.W., Hedetniemi, S.T., Slater, P.J.: Domination in Graphs: The Theory. Dekker, New York (1998) zbMATHGoogle Scholar
  17. 17.
    Haynes, T.W., Hedetniemi, S.T., Slater, P.J.: Domination in Graphs: Advanced Topics. Dekker, New York (1998) zbMATHGoogle Scholar
  18. 18.
    Haynes, T.W., Hedetniemi, S.M., Hedetniemi, S.T., Henning, M.A.: Domination in graphs applied to electric power networks. SIAM J. Discrete Math. 15(4), 519–529 (2002) MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Hon, W.-K., Liu, C.-S., Peng, S.-L., Tang, C.Y.: Power domination on block-cactus graphs. In: Proc. the 24th Workshop Combin. Math. and Comput. Theory, pp. 280–284 (2007) Google Scholar
  20. 20.
    Hsu, W.-L., Tsai, K.-H.: Linear time algorithms on circular-arc graphs. Inf. Process. Lett. 40(3), 123–129 (1991) MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Kneis, J., Mölle, D., Richter, S., Rossmanith, P.: Parameterized power domination complexity. Inf. Process. Lett. 98(4), 145–149 (2006) zbMATHCrossRefGoogle Scholar
  22. 22.
    Lee, D.T., Sarrafzadeh, M., Wu, Y.F.: Minimum cuts for circular-arc graphs. SIAM J. Comput. 19(6), 1041–1050 (1990) MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Liao, C.-S., Lee, D.T.: Power domination problem in graphs. In: Proc. the 11th International Comput. and Combin. Conference, pp. 818–828 (2005) Google Scholar
  24. 24.
    Lien, K.-P., Liu, C.-W., Yu, C.-S., Jiang, J.-A.: Transmission network fault location observability with minimal PMU placement. IEEE Trans. Power Deliv. 21(3), 1128–1136 (2006) CrossRefGoogle Scholar
  25. 25.
    Nuqui, R.F., Phadke, A.G.: Phasor measurement unit placement techniques for complete and incomplete observability. IEEE Trans. Power Deliv. 20(4), 2381–2388 (2005) CrossRefGoogle Scholar
  26. 26.
    Pai, K.-J., Chang, J.-M., Wang, Y.-L.: A simple algorithm for solving the power domination problem on grid graphs. In: Proc. the 24th Workshop Combin. Math. and Comput. Theory, pp. 256–260 (2007) Google Scholar
  27. 27.
    Pai, K.-J., Chang, J.-M., Wang, Y.-L.: Restricted power domination and fault-tolerant power domination on grids. Discrete Appl. Math. 158(10), 1079–1089 (2010) MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    Peng, J., Sun, Y., Wang, H.F.: Optimal PMU placement for full network observability using Tabu search algorithm. Int. J. Electr. Power Energy Syst. 28, 223–231 (2006) CrossRefGoogle Scholar
  29. 29.
    Phadke, A.G.: Synchronized phasor measurements in power systems. IEEE Comput. Applic. Power 6(2), 10–15 (1993) MathSciNetCrossRefGoogle Scholar
  30. 30.
    Phadke, A.G., Thorp, J.S., Nuqui, R.F., Zhou, M.: Recent developments in state estimation with phasor measurements. In: Proc. the IEEE Power Systems Conference and Exposition, pp. 1–7 (2009) CrossRefGoogle Scholar
  31. 31.
    Raible, D., Fernau, H.: Power domination in O (1.7548n) using reference search trees. In: Proc. the 19th International Symposium Algorithms and Computation, pp. 136–147 (2008) Google Scholar
  32. 32.
    Ramalingam, G., Pandu Rangan, C.: A unified approach to domination problems in interval graphs. Inf. Process. Lett. 27(5), 271–274 (1988) MathSciNetzbMATHCrossRefGoogle Scholar
  33. 33.
    Tarjan, R.E.: Efficiency of a good but not linear set union algorithm. J. ACM 22(2), 215–225 (1975) MathSciNetzbMATHCrossRefGoogle Scholar
  34. 34.
    Terzija, V., Valverde, G., Cai, D., Regulski, P., Madani, V., Fitch, J., Skok, S., Begovic, M.M., Phadke, A.G.: Wide-area monitoring, protection, and control of future electric power networks. Proc. IEEE 99(1), 80–93 (2011) CrossRefGoogle Scholar
  35. 35.
    Tsai, K.-H., Lee, D.T.: k-best cuts for circular-arc graphs. Algorithmica 18(2), 198–216 (1997) MathSciNetzbMATHCrossRefGoogle Scholar
  36. 36.
    Xu, G., Kang, L., Shan, E., Zhao, M.: Power domination in block graphs. Theor. Comput. Sci. 359, 299–305 (2006) MathSciNetzbMATHCrossRefGoogle Scholar
  37. 37.
    Zhao, M., Kang, L., Chang, G.J.: Power domination in graphs. Discrete Math. 306(15), 1812–1816 (2006) MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Department of Industrial Engineering and Engineering ManagementNational Tsing Hua UniversityHsinchuTaiwan
  2. 2.Department of Computer Science and EngineeringNational Chung Hsing UniversityTaichungTaiwan
  3. 3.Dept. of Computer Science and Information EngineeringNational Taiwan UniversityTaipeiTaiwan
  4. 4.Institute of Information ScienceAcademia SinicaNankang, TaipeiTaiwan

Personalised recommendations