Connected power domination in graphs

  • Boris BrimkovEmail author
  • Derek Mikesell
  • Logan Smith


The study of power domination in graphs arises from the problem of placing a minimum number of measurement devices in an electrical network while monitoring the entire network. A power dominating set of a graph is a set of vertices from which every vertex in the graph can be observed, following a set of rules for power system monitoring. In this paper, we study the problem of finding a minimum power dominating set which is connected; the cardinality of such a set is called the connected power domination number of the graph. We show that the connected power domination number of a graph is NP-hard to compute in general, but can be computed in linear time in cactus graphs and block graphs. We also give various structural results about connected power domination, including a cut vertex decomposition and a characterization of the effects of various vertex and edge operations on the connected power domination number. Finally, we present novel integer programming formulations for power domination, connected power domination, and power propagation time, and give computational results.


Connected power domination Power domination Cactus graph NP-complete Integer program Zero forcing 

Mathematics Subject Classification

05C69 05C57 94C15 



This work is supported by the National Science Foundation, under Grants CMMI-1634550, CMMI-1404864, and DMS-1720225, and by the Rice University Academy of Fellows.


  1. Aazami A (2008) Hardness results and approximation algorithms for some problems on graphs. Ph.D. thesis, Ph.D. thesis. University of WaterlooGoogle Scholar
  2. Aazami A (2010) Domination in graphs with bounded propagation: algorithms, formulations and hardness results. J Comb Optim 19(4):429–456MathSciNetzbMATHGoogle Scholar
  3. Aazami A, Stilp K (2009) Approximation algorithms and hardness for domination with propagation. SIAM J Discrete Math 23(3):1382–1399MathSciNetzbMATHGoogle Scholar
  4. AIM Special Work Group (2008) Zero forcing sets and the minimum rank of graphs. Linear Algebra Appl 428(7):1628–1648MathSciNetGoogle Scholar
  5. Akhlaghi S, Zhou N, Wu NE (2016) PMU placement for state estimation considering measurement redundancy and controlled islanding. In: Power and Energy Society General Meeting (PESGM), pp 1–5Google Scholar
  6. Aminifar F, Fotuhi-Firuzabad M, Shahidehpour M, Khodaei A (2011) Probabilistic multistage PMU placement in electric power systems. IEEE Trans Power Deliv 26(2):841–849Google Scholar
  7. Bader DA, Kappes A, Meyerhenke H, Sanders P, Schulz C, Wagner D (2014) Benchmarking for graph clustering and partitioning. In: Encyclopedia of social network analysis and mining. Springer, pp 73–82Google Scholar
  8. Baldwin T, Mili L, Boisen M, Adapa R (1993) Power system observability with minimal phasor measurement placement. IEEE Trans Power Syst 8(2):707–715Google Scholar
  9. Barioli F, Fallat S, Hogben L (2004) Computation of minimal rank and path cover number for certain graphs. Linear Algebra Appl 392:289–303MathSciNetzbMATHGoogle Scholar
  10. Barioli F, Fallat S, Hogben L (2005) On the difference between the maximum multiplicity and path cover number for tree-like graphs. Linear Algebra Appl 409:13–31MathSciNetzbMATHGoogle Scholar
  11. Benson KF, Ferrero D, Flagg M, Furst V, Hogben L, Vasilevska V, Wissman B (2018) Power domination and zero forcing for graph products. Aust J Comb 70(2):221–235zbMATHGoogle Scholar
  12. Bondy JA, Murty USR (1976) Graph theory with applications, vol 290. Macmillan, LondonzbMATHGoogle Scholar
  13. Bozeman C, Brimkov B, Erickson C, Ferrero D, Flagg M, Hogben L (2018) Restricted power domination and zero forcing problems. J Comb Optim, in pressGoogle Scholar
  14. Brimkov B, Fast CC, Hicks IV (2018) Computational approaches for zero forcing and related problems. Eur J Oper Res, in pressGoogle Scholar
  15. Brimkov B, Fast CC, Hicks IV (2017) Graphs with extremal connected forcing numbers. arXiv:1701.08500 Google Scholar
  16. Brimkov B, Hicks IV (2017) Complexity and computation of connected zero forcing. Discrete Appl Math 229:31–45MathSciNetzbMATHGoogle Scholar
  17. Brueni DJ (1993) Minimal PMU placement for graph observability: a decomposition approach. Ph.D. thesis, Virginia TechGoogle Scholar
  18. Brueni DJ, Heath LS (2005) The PMU placement problem. SIAM J Discrete Math 19(3):744–761MathSciNetzbMATHGoogle Scholar
  19. Burgarth D, Giovannetti V (2007) Full control by locally induced relaxation. Phys Rev Lett 99(10):100501Google Scholar
  20. Camby E, Cardinal J, Fiorini S, Schaudt O (2014) The price of connectivity for vertex cover. Discrete Math Theor Comput Sci 16:207–223MathSciNetzbMATHGoogle Scholar
  21. Caro Y, West DB, Yuster R (2000) Connected domination and spanning trees with many leaves. SIAM J Discrete Math 13(2):202–211MathSciNetzbMATHGoogle Scholar
  22. Chang GJ, Dorbec P, Montassier M, Raspaud A (2012) Generalized power domination of graphs. Discrete Appl Math 160(12):1691–1698MathSciNetzbMATHGoogle Scholar
  23. Chang GJ, Roussel N (2015) On the $k$-power domination of hypergraphs. J Comb Optim 30(4):1095–1106MathSciNetzbMATHGoogle Scholar
  24. Desormeaux WJ, Haynes TW, Henning MA (2013) Bounds on the connected domination number of a graph. Discrete Appl Math 161(18):2925–2931MathSciNetzbMATHGoogle Scholar
  25. Desrochers M, Laporte G (1991) Improvements and extensions to the Miller–Tucker–Zemlin subtour elimination constraints. Oper Res Lett 10(1):27–36MathSciNetzbMATHGoogle Scholar
  26. Dilkina BN, Gomes CP (2010) Solving connected subgraph problems in wildlife conservation. In: CPAIOR, vol 6140. Springer, pp 102–116Google Scholar
  27. Dorbec P, Klavžar S (2014) Generalized power domination: propagation radius and Sierpiński graphs. Acta Appl Math 134(1):75–86MathSciNetzbMATHGoogle Scholar
  28. Edholm CJ, Hogben L, LaGrange J, Row DD (2012) Vertex and edge spread of zero forcing number, maximum nullity, and minimum rank of a graph. Linear Algebra Appl 436(12):4352–4372MathSciNetzbMATHGoogle Scholar
  29. Fan N, Watson J-P (2012) Solving the connected dominating set problem and power dominating set problem by integer programming. In: International conference on combinatorial optimization and applications. Springer, pp 371–383Google Scholar
  30. Ferrero D, Hogben L, Kenter FH, Young M (2017) Note on power propagation time and lower bounds for the power domination number. J Comb Optim 34(3):736–741MathSciNetzbMATHGoogle Scholar
  31. Fomin FV, Grandoni F, Kratsch D (2008) Solving connected dominating set faster than $2^n$. Algorithmica 52(2):153–166MathSciNetzbMATHGoogle Scholar
  32. Garey MR, Johnson DS (1979) Computers and intractability. W.H. Freeman & Co., San FranciscozbMATHGoogle Scholar
  33. Guo J, Niedermeier R, Raible D (2005) Improved algorithms and complexity results for power domination in graphs. In: FCT, vol 3623. Springer, pp 172–184Google Scholar
  34. Haynes TW, Hedetniemi SM, Hedetniemi ST, Henning MA (2002) Domination in graphs applied to electric power networks. SIAM J Discrete Math 15(4):519–529MathSciNetzbMATHGoogle Scholar
  35. Huang L-H, Chang GJ, Yeh H-G (2010) On minimum rank and zero forcing sets of a graph. Linear Algebra Appl 432(11):2961–2973MathSciNetzbMATHGoogle Scholar
  36. Huang L, Sun Y, Xu J, Gao W, Zhang J, Wu Z (2014) Optimal PMU placement considering controlled islanding of power system. IEEE Trans Power Syst 29(2):742–755Google Scholar
  37. IEEE reliability test data (2012).
  38. Kneis J, Mölle D, Richter S, Rossmanith P (2006) Parameterized power domination complexity. Inf Process Lett 98(4):145–149MathSciNetzbMATHGoogle Scholar
  39. Li Q, Cui T, Weng Y, Negi R, Franchetti F, Ilic MD (2013) An information-theoretic approach to PMU placement in electric power systems. IEEE Trans Smart Grid 4(1):446–456Google Scholar
  40. Li Y, Yang Z, Wang W (2017) Complexity and algorithms for the connected vertex cover problem in 4-regular graphs. Appl Math Comput 301:107–114MathSciNetGoogle Scholar
  41. Liao C-S (2016) Power domination with bounded time constraints. J Comb Optim 31(2):725–742MathSciNetzbMATHGoogle Scholar
  42. Liao C-S, Lee D-T (2005) Power domination problem in graphs. In: COCOON. Springer, pp 818–828Google Scholar
  43. Manousakis NM, Korres GN, Georgilakis PS (2012) Taxonomy of PMU placement methodologies. IEEE Trans Power Syst 27(2):1070–1077Google Scholar
  44. Martin RK (1991) Using separation algorithms to generate mixed integer model reformulations. Oper Res Lett 10(3):119–128MathSciNetzbMATHGoogle Scholar
  45. Mili L, Baldwin T, Phadke A (1991) Phasor measurements for voltage and transient stability monitoring and control. In: Workshop on application of advanced mathematics to power systems, San FranciscoGoogle Scholar
  46. Miller CE, Tucker AW, Zemlin RA (1960) Integer programming formulation of traveling salesman problems. J ACM 7(4):326–329MathSciNetzbMATHGoogle Scholar
  47. Nylen PM (1996) Minimum-rank matrices with prescribed graph. Linear Algebra Appl 248:303–316MathSciNetzbMATHGoogle Scholar
  48. Peng J, Sun Y, Wang H (2006) Optimal PMU placement for full network observability using Tabu search algorithm. Int J Electr Power Energy Syst 28(4):223–231Google Scholar
  49. Quintão FP, da Cunha AS, Mateus GR, Lucena A (2010) The k-cardinality tree problem: reformulations and Lagrangian relaxation. Discrete Appl Math 158(12):1305–1314MathSciNetzbMATHGoogle Scholar
  50. Row D (2012) Zero forcing number, path cover number, and maximum nullity of cacti. Involve 4(3):277–291MathSciNetzbMATHGoogle Scholar
  51. Sampathkumar E, Walikar HB (1979) The connected domination number of a graph. J Math Phys Sci 13(6):607–613MathSciNetzbMATHGoogle Scholar
  52. Sodhi R, Srivastava S, Singh S (2011) Multi-criteria decision-making approach for multi-stage optimal placement of phasor measurement units. IET Gener Transm Distrib 5(2):181–190Google Scholar
  53. Tarjan RE (1974) A note on finding the bridges of a graph. Inf Process Lett 2(6):160–161MathSciNetzbMATHGoogle Scholar
  54. Yang B (2013) Fast-mixed searching and related problems on graphs. Theor Comput Sci 507:100–113MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Computational and Applied MathematicsRice UniversityHoustonUSA

Personalised recommendations