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Note on Dominating Set Problems

Abstract

This paper focuses on connected dominating set problems on a network: basic optimization formulations of problems, multicriteria problem formulations, and problem formulation with multiset estimate. A survey of the literature on problems and solution schemes is presented. Numerical examples illustrate connected dominating set problems. New integer programming formulations of dominating set problems (multicriteria problems, problems with multiset estimates) are proposed.

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REFERENCES

  1. F. N. Abu-Khzam, A. E. Mouawad, and M. Liedloff, “An exact algorithm for connected red-blue dominating set,” J. Discr. Alg. 9, 252–262 (2011).

    MathSciNet  MATH  Google Scholar 

  2. C. Adjih, P. Jacquet, and L. Viennot, “Computing connected dominating sets with multipoint relays,” Ad Hoc & Sensor Wir. Netw. (Mar.), 27–39 (2005).

  3. J. A. Torkestani and M. R. Meybodi, “Clustering the wireless Ad Hoc networks: distributed learning automata approach,” J. Parallel Distr. Comput. 70, 394–405 (2010).

    MATH  Google Scholar 

  4. J. A. Torkestani and M. R. Meybodi, “Weighted Steiner connected dominating set and its application to multicast routing in wireless MANETs,” Wir. Pers. Commun. 60 (2), 145–169 (2011).

    Google Scholar 

  5. J. A. Torkestani, “An adaptive backbone formation algorithm for wireless sensor networks,” Comp. Commun. 35, 1333–1344 (2012).

    Google Scholar 

  6. J. A. Torkestani, “Algorithms for Steiner connected dominating set problem based on learning automata theory,” Int. J. Foundat. Comp. Sci. 26 (6), 769–801 (2015).

    MathSciNet  MATH  Google Scholar 

  7. R. B. Allan, R. Laskar, and S. T. Hedetniemi, “A note on total domination,” Discr. Math. 49 (1), 7–13 (1984).

    MathSciNet  MATH  Google Scholar 

  8. J. Alber, H. Fan, M. R. Fellows, R. Niedereier, F. A. Rosamond, and U. Stege, “A refined search tree technique for dominating set on planar graphs,” J. Comput. Syst. Sci. 71 (4), 385–405 (2005).

    MathSciNet  MATH  Google Scholar 

  9. M. Albuquerque and T. Vidal, http://arxiv.org/ abs/1808.09809 [cs.AI].

  10. N. Alon, F. Fomin, G. Gutin, M. Krivelevich, and S. Saurabh, “Spanning directed trees with many leaves,” SIAM J. Discr. Math. 23 (1), 466–476 (2009).

    MathSciNet  MATH  Google Scholar 

  11. N. Alon and S. Gutner, “Linear time algorithms for finding a dominating set of fixed size in degenerated graphs,” Algorithmica 54, 544–556 (2009).

    MathSciNet  MATH  Google Scholar 

  12. J. D. Alvarado, S. Dantas, E. Mohr, and D. Rautenbach, “On the maximum number of minimum dominating sets in forests,” Discr. Math. 342, 934–942 (2019).

    MathSciNet  MATH  Google Scholar 

  13. K. M. Alzoubi, P.-J. Wan, and O. Frieder, “Maximal independent set, weakly connected dominating set, and induced spanners for mobile ad-hoc networks,” Int. J. Foundat. Comp. Sci. 14, 287–303 (2003).

    MATH  Google Scholar 

  14. C. Ambuhl, T. Erlebach, M. Mihalak, and M. Nunkesser, “Constant-factor approximation for minimum-weight (connected) dominating sets in unit disk graph,” in APPROX-RANDOM 2006, LNCS 4110 (Springer, 2006), pp. 3–14.

    Google Scholar 

  15. D. V. Andrade, M. G. C. Resende, and R. F. Werneck, “Fast local search for the maximum independent set problem” J. of Heur. 18, 525–547 (2012).

    MATH  Google Scholar 

  16. X. Bai, D. Zhao, S. Bai, Q. Wang, W. Li, and D. Mu, “Minimum connected dominating sets in heterogeneous 3D wireless Ad Hoc networks,” Ad Hoc Netw. 97, art. 102023 (2020).

    Google Scholar 

  17. A. Berger, T. Fukunaga, H. Nagamochi, and O. Parekh, “Approximability of the capacitated b‑edge dominating set problem,” Theor. Comp. Sci. 385 (1–3), 202–213 (2007).

    MathSciNet  MATH  Google Scholar 

  18. A. Berger and O. Parekh, “Linear time algorithms for generalized edge dominating set problems,” Algorithmica 59, 244–254 (2008).

    MathSciNet  MATH  Google Scholar 

  19. S. Bermudo, J. C. Hernandez-Gomez, and J. M. Sigarreta, “Total k-domination in strong product graphs,” Discr. Appl. Math. 263, 51–58 (2019).

    MathSciNet  MATH  Google Scholar 

  20. S. Bermudo, A. C. Martinez, MiraF. A. Hernandez, and J. M. Sigarreta, “On the global total k-domination number of graphs,” Discr. Appl. Math. 263, 42–50 (2019).

    Google Scholar 

  21. J. Blum, M. Ding, A. Thaeler, and X. Cheng, “Connected dominating set in sensor networks and MANETs,” in Handbook of Combinatorial Optimization, by Ed. D.-Z. Du and P. M. Pardalos, (Springer, 2005), pp. 329–369.

  22. A. Buchanan, J. S. Sung, V. Boginski, and S. Butenko, “On connected dominating set of restricted diameter,” EJOR 236 (2), 410–418 (2014).

    MathSciNet  MATH  Google Scholar 

  23. S. Butenko, X. Cheng, C. A. S. Oliveira, and P. M. Pardalos, “A new heuristic for the minimum connected dominating set problem on ad hoc wireless networks” in Recent Developments in Cooperative Control and Optimization (Springer, 2004), pp. 61–73.

    Google Scholar 

  24. Y. Caro, D. B. West, and R. Yuster, “Connected domination and spanning trees with many leaves,” SIAM J. Discr. Math. 13 (2), 202–211 (2000).

    MathSciNet  MATH  Google Scholar 

  25. Y. Caro, A. Hansberg, and M. Henning, “Fair domination in graphs,” Discr. Math. 312, 2905–2914 (2012).

    MathSciNet  MATH  Google Scholar 

  26. R. Carr, T. Fujito, G. Konjevod, and O. Parekh, “A, 2 1/10-approximation algorithm for a generalization of the weighted edge-dominating set problem,” J. Comb. Optim. 5, 317–326 (2001).

    MathSciNet  MATH  Google Scholar 

  27. M.-S. Chang, Weighted domination of cocomparability graphs. Discr. Appl. Math. 80, 135–148 (1997).

    MathSciNet  MATH  Google Scholar 

  28. Y. P. Chen and A. L. Liestman, “Approximating minimum size weakly-connected dominating sets for clus-tering mobile ad hoc networks,” MobiHoc, 165–172, (2002).

  29. Y. P. Chen and A. L. Liestman, “Maintaining weakly connected dominating sets for clustering Ad-Hoc networks,” Ad Hoc Netw. 3, 629–642 (2005).

    Google Scholar 

  30. X. Cheng, X. Huang, D. Li, W. Wu, and D.-Z. Du, “A polynomial-time approximation scheme for minimum connected dominating set in ad hoc wireless networks,” Networks 42 (4), 202–208 (2003).

    MathSciNet  MATH  Google Scholar 

  31. C. J. Cheng, C. Lu, and Y. Zhou, “The k-power domination problem in weighted trees,” in AAIM 2018, LNCS 11343 (Springer, 2018), pp. 149–160.

    Google Scholar 

  32. M. Chlebik and J. Chlebikova, “Approximation hardness of edge dominating set problems,” J. Comb. Optim. 11 (3), 279–290 (2006).

    MathSciNet  MATH  Google Scholar 

  33. E. J. Cockayne, R. Dawes, and S. T. Hedetniemi, “Total domination in graphs. Networks,” 10, 211–215 (1980).

    MathSciNet  MATH  Google Scholar 

  34. R. S. Coelho, P. F. S. Moura, and Y. Wakabayashi, “The k-hop connected dominating set problem: approximation and hardness.” J. Comb. Optim. 34, 1060–1083 (2017).

    MathSciNet  MATH  Google Scholar 

  35. J.-F. Couturier, P. Heggernes, van 't P. Hof, and D. Kratsch, “Minimal dominating sets in graph classes: Combinatorial bounds and enumeration. Theor. Comp. Sci. 487, 82–94 (2013).

    Google Scholar 

  36. Z. A. Dagdeviren, D. Aydin, and M. Cinsdikici, “Two population-based optimization algorithms for minimum weight connected dominating set problem,” Appl. Soft Comput. 59, 644–658 (2017).

    Google Scholar 

  37. F. Dai and J. Wu, “An extended localized algorithm for connected dominating set formation in Ad Hoc wireless networks,” IEEE Trans. Parallel & Distrib. Syst. 15, 908–920 (2004).

    Google Scholar 

  38. F. Dai and J. Wu, “On constructing k-connected k‑dominating set in wireless ad hoc and sensor networks,” J. Parallel & Distr. Comput. 66, 947–958 (2006).

    MATH  Google Scholar 

  39. T. N. Dinh, Y. Shen, D. T. Nguyen, and M. T. Thai, “On the approximability of positive influence dominating set in social networks.” J. Com. Optim. 27, 487–503 (2014).

    MathSciNet  MATH  Google Scholar 

  40. M. Dom, D. Lokshtanov, S. Saurabh, and Y. Villanger, “Capacitated domination and covering: a parameterized perspective,” in Proc. 3rd IWPEC, LNCS 5018 (Springer, 2008), pp. 78–90.

  41. M. Dorfling and M. A. Henning, “A note on power domination in grid graphs,” Discr. Appl. Math. 154, 1023–1027 (2006).

    MathSciNet  MATH  Google Scholar 

  42. D.-Z. Du, M. T. Thai, Y. Li, D. Liu, and S. Zhu, “Strongly connected dominating sets in wireless sensor networks with unidirectional links,” in APWeb 2006, LNCS 3841 (Springer, 2006), pp. 13–24.

    Google Scholar 

  43. D.-Z. Du and P.-J. Wan, Connected Dominating Set: Theory and Applications (Springer, 2013).

    MATH  Google Scholar 

  44. H. Du, Q. Ye, J. Zhong, Y. Wang, W. Lee, and H. Park, “PTAS for minimum connected dominating set with routing cost constraint in wireless sensor networksin,” COCOA 2010, Part 1, LNCS 6508 (Springer, 2020), pp. 252–259.

  45. H. Du, Q. Ye, J. Zhong, Y. Wang, W. Lee, and H. Park, “Polynomial-time approximation scheme for minimum connected dominating set under routing cost constraint in wireless sensor networks,” Theor. Comp. Sci. 447, 38–43 (2012).

    MathSciNet  MATH  Google Scholar 

  46. H. Du, L. Ding, W. Wu, D. Kim, P. M. Pardalos, and J. Willson, “Connected dominating set in wireless networks,” in Handbook of Combinatorial Optimization, Ed. by P. M. Pardalos, R. L. Graham, and D.-Z. Du, 2nd ed., (Springer, 2013), pp. 783–834.

    Google Scholar 

  47. H. Du and H. Luo, “Routing-cost constrained connected dominating set,” in M.Y. Kao (ed.), Encyclopedia of Algorithms, Ed. by M. Y. Kao, (Springer, 2016), pp. 1879–1883.

  48. K. Erciyes, O. Dagdeviren, D. Cokeslu, and D. Ozsoyeller, “Graph theoretic clustering algorithms in mobile ad hoc networks and wireless sensor networks - survey,” Appl. Comput. Math. 6 (2), 162–180 (2007).

    MathSciNet  MATH  Google Scholar 

  49. F. V. Fomin, D. Kratsch, and G. J. Woeginger, “Exact (exponential) algorithms for the dominating set problem” in LNCS 3353, Ed. by J. Hromkovic, M. Nagl, and B. Westfechtel (Springer, 2004), pp. 245–256.

  50. F. V. Fomin and D. M. Thilikos, “Dominating sets in planar graphs: branch-width and exponential speed-up,” SIAM J. Comput. 36 (2), 281–309 (2006).

    MathSciNet  MATH  Google Scholar 

  51. D. Fu, L. Han, L. Liu, Q. Gao, and Z. Feng, “An efficient centralized algorithm for connected dominating set on wireless networks,” Procedia CS 56, 162–167 (2015).

    Google Scholar 

  52. T. Fujito, “Approximability of the independent/connected edge dominating set problems,” Inform. Proc. Lett. 79, 261–266 (2001).

    MathSciNet  MATH  Google Scholar 

  53. T. Fujito and H. Nagamochi, “A 2-approximation algorithm for the minimum weight edge dominating set problem,” Discr. Appl. Math. 118 (3), 199–207 (2002).

    MathSciNet  MATH  Google Scholar 

  54. T. Fujie, “An exact algorithm for the maximum leaf spanning tree problem,” Comp. and Oper. Res. 30, 1931–1944 (2003).

    MathSciNet  MATH  Google Scholar 

  55. T. Fukunaga and H. Nagamochi, “Approximation algorithm for the b-edge dominating set problem and its related problems,” in COCOON 2005, LNCS 3595 (Springer, 2005), pp. 747–756.

    Google Scholar 

  56. T. Fukunaga, Approximation algorithms for highly connected multi-dominating sets in unit disk graphs. Algorithmica 80 (11), 3270–3292 (2018).

    MathSciNet  MATH  Google Scholar 

  57. T. Fukunaga, “Adaptive algorithms for finding connected dominating sets in uncertain graphs,” Electr. Prepr., 19 p., Dec 29, (2019). http://arxiv.org/ abs/1912.12665 [cs.DS]

  58. S. Funke, A. Kesselman, U. Meyer, and M. Segal, “A simple improved distributed algorithm for minimum CDS in unit disk graphs,” ACM Trans. Sensor Netw. 2 (3), 444–453 (2006).

    Google Scholar 

  59. X. Gao, W. Wag, Z. Zhang, S. Zhu, and W. Wu, “A PTAS for minimum d-hop connected dominating set in growth-bounded graphs,” Optim. Lett. 4, 321–333 (2010).

    MathSciNet  MATH  Google Scholar 

  60. M. R. Garey and D. S. Johnson, Computers and Intractability. The Guide to the Theory of NP-Completeness (W. H. Freeman and Company, San Francisco, 1979).

    MATH  Google Scholar 

  61. W. Goddard and J. Lyle, “Independent dominating sets in triangle-free graphs,” J. Comb. Optim. 23 (1), 9–20 (2012).

    MathSciNet  MATH  Google Scholar 

  62. S. Guha and S. Khuller, “Approximation algorithms for connected dominating sets,” Algorithmica 20, 374–387 (1998).

    MathSciNet  MATH  Google Scholar 

  63. M. Hajian and N. J. Rad, “A new lower bound on the double domination number of a graph,” Discr. Appl. Math. 254, 280–282 (2019).

    MathSciNet  MATH  Google Scholar 

  64. J. Harant and M. A. Henning, “On double dominating in graphs,” Discussiones Math. 25, 29–34 (2005).

    MATH  Google Scholar 

  65. F. Harary and T. W. Haynes, “Double domination in graphs,” Ars Combin. 55, 201–213 (2000).

    MathSciNet  MATH  Google Scholar 

  66. T. W. Haynes, S. T. Hedetniemi, and P. J. Slater, Fundamentals of Domination in Graphs (Marcel Dekker, 1998).

    MATH  Google Scholar 

  67. T. W. Haynes, S. M. Hedetniemi, S. T. Hedetniemi, and M. A. Henning, “Domination in graphs applied to electrical power networks,” SIAM J. on Discr. Math. 15, 519–529 (2002).

    MathSciNet  MATH  Google Scholar 

  68. J. He, S. Ji, P. Fan, Y. Pan, and Y. Li, in “Constructing a load-balanced virtual backbone in wireless sensor networks,” in Proc. 2012 Int. Conf. on Computing, Networking and Communication (ICNC), 2012, pp. 959–963.

  69. A.-R. Hedar and R. Ismail, “Hybrid genetic algorithm for minimum dominating set problem,” in ICCSA 2010, pp. 457–467.

  70. M. A. Henning and N. J. Rad, “Locating-total domination in graphs,” Discr. Appl. Math. 160, 1986–1993 (2012).

    MathSciNet  MATH  Google Scholar 

  71. M. A. Henning and N. J. Rad, “Bounds on neighborhood total domination in graphs,” Discr. Appl. Math. 161, 2460–2466 (2013).

    MathSciNet  MATH  Google Scholar 

  72. M. A. Henning and A. Yeo, Total Domination in Graphs (Springer, 2013).

    MATH  Google Scholar 

  73. M. A. Henning and A. J. Marcon, “On matching and semitotal domination in graphs,” Discr. Math. 324, 13–18 (2014).

    MathSciNet  MATH  Google Scholar 

  74. M. A. Henning and D. Pradhan, “Algorithmic aspects of upper paired-domination in graphs,” Theor. Comp. Sci. 804, 98–114 (2020).

    MathSciNet  MATH  Google Scholar 

  75. M. A. Henning, S. Pal, and D. Pradhan, “Algorithm and hardness results on hop domination in graphs,” Inform. Proc. Lett. 153, 105872 (2020).

    MathSciNet  MATH  Google Scholar 

  76. N. Hjuler, G. F. Italiano, N. Parotsidis, and D. Saulpic, “Dominating sets and connected dominating sets in dynamic graphs,” in STACS 2019, pp. 35:1–35:17.

  77. C. K. Ho, Y. P. Singh, and H. T. Ewe, “An enhanced ant colony optimization metaheuristic for the minimum dominating set problem,” Appl. Artif. Intell. 20 (10), 881–903 (2006).

    Google Scholar 

  78. J. Horton and K. Kilakos, “Minimum edge dominating sets,” SIAM J. Discr. Math. 6 (3), 375–387 (1993).

    MathSciNet  MATH  Google Scholar 

  79. R. W. Irving, “On approximating the minimum independent dominating set,” Inf. Proc. Lett. 37 (4), 197–200 (1991).

    MathSciNet  MATH  Google Scholar 

  80. L. Jia, R. Rajaraman, and T. Suel, “An efficient distributed algorithm for constructing small dominating sets,” Distrib. Comput. 15 (4), 193–205 (2002).

    MATH  Google Scholar 

  81. R. K. Jullu, P. R. Prasad, and G. K. Das, “Distributed construciton of connected dominating set in unit disk graphs,” J. Parallel and Distr. Comput. 104, 159–166 (2017).

    Google Scholar 

  82. M. J. Kao, C. S. Liao, and D. T. Lee, “Capacitated domination problem,” Algorithmica 60 (2), 274–300 (2011).

    MathSciNet  MATH  Google Scholar 

  83. D. J. Kleitman and D. B. West, “Spanning trees with many leaves,” SIAM J. Discr. Math. 4 (1), 99–106 (1991).

    MathSciNet  MATH  Google Scholar 

  84. S. Kundu and S. Majumder, “A linear time algorithm for optimal k-hop dominating set of a tree,” Inf. Process. Lett. 116 (2), 197–202 (2016).

    MathSciNet  MATH  Google Scholar 

  85. J. K. Lan and G. J. Chang, “On the mixed domination problem in graphs,” Theor. Comp. Sci. 476, 84–93 (2013).

    MathSciNet  MATH  Google Scholar 

  86. E. Lappas, S. D. Nikolopoulos, and L. Palios, “An O(n)-time algorithm for paired-domination on permutation graphs,” Eur. J. Combin. 34 (3), 593–608 (2013).

    MathSciNet  MATH  Google Scholar 

  87. M. Sh. Levin, Modular System Design and Evaluation (Sprigner, 2015).

    Google Scholar 

  88. M. Sh. Levin, “On combinatorial optimization for dominating sets (literature survey, new models),” Preprint (ResearchGate)), Sep. 4, (2020). Concurently: arxiv 2009.09288.https://doi.org/10.13140/RG.2.2.34919.68006

  89. Y. Li, Y. Wu, C. Ai, and F. Beyah, “On the construction of k-connected m-dominating sets in wireless networks,” J. Comb. Optim. 23 (1), 118–139 (2012).

    MathSciNet  MATH  Google Scholar 

  90. H. Li, Y. Yang, and B. Wu, “2-edge connected dominating sets and 2-connected dominating sets of a graph,” J. Comb. Optim. 31 (2), 713–724 (2016).

    MathSciNet  MATH  Google Scholar 

  91. D. Liang, Z. Zhang, X. Liu, W. Wang, and Y. Jiang, “Approximation algorithms for minimum weight partial connected set cover problem,” J. Comb. Optim. 31 (2), 696–712 (2016).

    MathSciNet  MATH  Google Scholar 

  92. C.-S. Liao, T.-J. Hsieh, X.-C. Guo, and C.-C. Chu, “Hybrid search for the optimal pmu placement problem on a power grid,” EJOR 243 (3), 985–994 (2015).

    MathSciNet  MATH  Google Scholar 

  93. M. Liedloff, I. Todinca, and Y. Villanger, “Solving capacitated dominating set by using covering by subsets and maximum matching,” Discr. Appl. Math. 168, 60–68 (2014).

    MathSciNet  MATH  Google Scholar 

  94. Z. Lin, H. Liu, X. Chu, Y.-W. Leung, and I. Stojmenovic, “Maximizing lifetime of connected-dominating set in cognitive radio,” in NETWORKING 2012, Part II, LNCS 7290 (Springer, 2012), pp. 316–330.

    Google Scholar 

  95. G. Lin, W. Zhu, and M. M. Ali, “An effective hybrid memetic algorithm for the minimum weight dominating set problem,” IEEE Trans. on Evolut. Comput. 20 (6), 892–907 (2016).

    Google Scholar 

  96. G. Lin, J. Guan, and H. Feng, “An ILP based memetic algorithm for finding positive influence dominating sets in social networks,” Physica A 500, 199–209 (2018).

    MathSciNet  Google Scholar 

  97. C.-H. Liu, S.-H. Poon, and J.-Y. Lin, “Independent dominating set problem revised,” Theor. Comp. Sci. 562, 1–22 (2015).

    MATH  Google Scholar 

  98. D. Lokshtanov, M. Mnich, and S. Saurabh, “A linear kernel for planar connected dominating set,” Theor. Comp. Sci. 412, 2536–2543 (2011).

    MathSciNet  MATH  Google Scholar 

  99. C. Luo, W. Chen, J. Yu, Y. Wang, and D. Li, “A novel centralized algorithm for constructing virtual back-bones in wireless sensor networks,” EURASIP J. Wir. Commun. and Netw., art. 55 (2018).

  100. M. Min, H. Du, X. Jia, C. X. Huang, S. C.-H. Huang, and W. Wu, “Improving construction for connected dominating set with Steiner tree in Wireless Sensor Networks,” J. Glob. Optim. 35, 111–119 (2006).

    MathSciNet  MATH  Google Scholar 

  101. J. P. Mohanty, C. Mandal, C. Reade, and A. Das, “Construction of minimum connected dominating set in wireless sensor networks,” Ad Hoc Netw. 42, 61–73 (2016).

    Google Scholar 

  102. J. P. Mohanty, C. Mandal, and C. Reade, “Distributed construction of minimum Connected Dominaitng Set in wireless sensor network using two-hop information,” Comp. Netw. 123, 137–152 (2017).

    Google Scholar 

  103. T. N. Nguen and D. T. Huynh, “Connected d-hop dominating sets in mobile ad hoc networks,” in Proc. 2005 4th Int. Symp. on Modeling and Optimization in Mobile, Ad Hoc and Wireless Networks, 2006, Vols. 1 and 2.

  104. T. Nieberg and J. Hurink, “A PTAS for the minimum dominating set problem in unit disk graphs,” in WAOA 2005, LNCS 3879 (Springer, 2005), pp. 296–306,

    Google Scholar 

  105. F. G. Noccetti, J. S. Gonzalez, and I. Stojmenovic, “Connectivity based k-hop clustering in wireless ad hoc networks,” Telecom. Syst. 22 (1-4), 205–220 (2003).

    Google Scholar 

  106. Z. Nutov, “Improved approximation algorithms for k‑connected m-dominating set problems,” Electr. Prepr., 6 p., Mar. 13, (2017). http://arxiv.org/abs/ 1703.04230 [cs.DC].

  107. C. A. S. Oliveira and P. M. Pardalos, “Ad Hoc networks: optimization problems and solution methods,” in M. X. Cheng, Y. Li, and D.-Z. Du (eds), Combinatorial Optimization in Communication Networks (Springer, 2006), pp. 147–170.

    Google Scholar 

  108. B. S. Panda and D. Pradhan, “A linear time algorithm for computing a minimum paired-dominating set of a convex bipartite graph,” Discr. Appl. Math. 161, 1776–1783 (2013).

    MathSciNet  MATH  Google Scholar 

  109. N. Parthiban, I. Rajasingh, and Rajan R. Sundara, “Minimum connected dominating set for certain circulant networks,” Procedia CS 57, 587–591 (2015).

    Google Scholar 

  110. P. Pinacho-Davidson, S. Bouamama, and C. Blum, “Application of CMSA to the minimum capacitated dominating set problem,” in GECCO 2019, pp. 321–328.

  111. A. Potluri and A. Singh, “Hybrid metaheuristic algorithms for minimum weight dominating set,” Appl. Soft Comput. 13, 76–88 (2013).

    Google Scholar 

  112. D. Pradhan and B. S. Panda, “Computing a minimum paired-dominating set in strongly orderable graphs,” Discr. Appl. Math. 253, 37–50 (2019).

    MathSciNet  MATH  Google Scholar 

  113. H. Qiao, L. Kang, M. Gardei, and D.-Z. Du, “Paired-domination of trees,” J. Glob. Optim. 25 (1), 43–54 (2003).

    MathSciNet  MATH  Google Scholar 

  114. N. J. Rad and L. Volkmann, “A note on the independent domination number in graphs,” Discr. Appl. Math. 161, 3087–3089 (2013).

    MathSciNet  MATH  Google Scholar 

  115. R. Ramalakshmi and S. Radhaktishnan, “Energy efficient stable connected dominating set construction in mobile ad hoc networks,” in CCSIT 2012, Part I, LNICST 84 (Springer, 2012), pp. 64–72, 2012.

    Google Scholar 

  116. J. M. M. van Rooij and H. L. Bodlaender, “Exact algorithms for dominating set,” Discr. Appl. Math. 159, 2147–2164 (2011).

    MathSciNet  MATH  Google Scholar 

  117. L. Ruan, H. Du, X. Jia, W. Wu, Y. Li, and K.-I. Ko, “A greedy approximation for minimum connected dominating sets,” Theor. Comp. Sci. 329 (1-3), 325–330 (2004).

    MathSciNet  MATH  Google Scholar 

  118. O. Schaudt and R. Schrader, “The complexity of connected dominating sets and total dominating sets with specified induced subgraphs,” Inf. Proc. Lett. 112, 953–957 (2012).

    MathSciNet  MATH  Google Scholar 

  119. W. Shang, F. Yao, P. Wan, and X. Hu, “On minimum m-connected k-dominating set problem in unit disc graph,” J. of Comb. Optim. 16 (2), 99–106 (2008).

    MathSciNet  MATH  Google Scholar 

  120. T. Shi, S. Cheng, Z. Cai, Y. Li, and J. Li, “Exploiting connected dominating sets in energy harvest networks,” IEEE/ACM Trans. on Netw. 25 (3), 1803–1817 (2017).

    Google Scholar 

  121. Y. Shi, Z. Zhang, and D.-Z. Du, “Approximation algorithm for minimum weight (k; m)-CDS problem in unit disk graph,” Electr. Prepr., Jan. 4, 2019. http://arxiv.org/abs/1508.005515 [cs.DM].

  122. L. Simonetti, A. S. da Cunha, and A. Lucena, “The minimum connected dominating set problem: formulation, valid inequalities and a Branch-and-Bound algorithm,” in INOC 2011, LNCS 6701 (Springer, 2011), pp. 162–169.

    Google Scholar 

  123. I. Stojmenovic, M. Seddigh, and J. Zunic, “Dominating sets and neighbor elimination-based broadcasting algorithms in wireless networks,” IEEE Trans. Paral. and Distr. Syst. 13, 14–25 (2002).

    Google Scholar 

  124. X. Sun, Y. Yang, and M. Ma, “Minimum connected dominating set algorithms for Ad Hoc networks,” Sensors 19 (8), art. 1919 (2019).

    Google Scholar 

  125. S. Surendran and S. Vijayan, “Distributed computation of connected dominating set for multi-hop wireless networks,” Procedia CS 63, 482–487 (2015).

    Google Scholar 

  126. A. Suzuki, A. E. Mouawad, and N. Nishimura, “Reconfiguration of dominating sets,” J. Comb. Optim. 32 (4), 1182–1195 (2016).

    MathSciNet  MATH  Google Scholar 

  127. M. Thai, N. Zhang, R. Tiwari, and X. Xu, “On approximation algorithms of k-connected m-dominating sets in disk graphs,” Theor. Comput. Sci. 385 (1–3), 49–59 (2007).

    MathSciNet  MATH  Google Scholar 

  128. Y. T. Tsai, Y. L. Lin, and F. R. Hsu, “Efficient algorithms for the minimum connected domination on trapezoid graphs,” Inform. Sci. 177 (12), 2405–2417 (2007).

    MathSciNet  MATH  Google Scholar 

  129. F. J. Vazquez-Araujo, A. Dapena, M. J. S. Salorio, and P.-M. Castro-Castro, “Calculation of the connected dominating set considering vertex importance metrics,” Entropy 20 (2) (2018).

  130. P.-J. Wan and K. M. Alzoubi, “A simple heuristic for minimum connected dominating set in graphs,” Int. J. of Found. Comp. Sci. 14 (2), 323–333 (2003).

    MathSciNet  MATH  Google Scholar 

  131. P.-J. Wan, L. Wang, and F. Yao, “Two-phase approximation algorithms for minimum CDS in wireless ad hoc networks,” in IEEE ICDCS, (IEEE, New York, 2008), pp. 337–344.

    Google Scholar 

  132. F. Wang, E. Camacho, and K. Xu, “Positive influence dominating set in social networks,” Theor. Comp. Sci. 412 (3), 265–269 (2011).

    MathSciNet  MATH  Google Scholar 

  133. Z. Wang, W. Wang, J.-M. Kim, B. Thuraisingham, and W. Wu, “PTAS for the minimum weighted dominating set in growth bounded graphs,” J. Glob. Optim. 54 (3), 641–648 (2012).

    MathSciNet  MATH  Google Scholar 

  134. Y. Wang, W. Wang, and X. -Y. Li, “Weighted connected dominating set,” in Kao M.-Y. (ed), Encyclopedia of Algorithms (Springer, 2016), pp. 2359–2363.

    Google Scholar 

  135. J. Wu and H. Li, “A dominating set based routing scheme in Ad Hoc wireless sensor networks,” Telecom. Syst. 18 (1-3), 13–36 (2001).

    MATH  Google Scholar 

  136. J. Wu and W. Lou, “Extended multipoint relays to determine connected dominating sets in MANETs,” IEEE Trans. on Comput. 55, 334–347 (2006).

    Google Scholar 

  137. Y.-F. Wu, Y.-L. Xu, and G.-L. Chen, “Approximation algorithms for Steiner connected dominating set,” J. Comp. Sci. and Techn. 20 (5), 713–716 (2005).

    MathSciNet  Google Scholar 

  138. W. Wu, H. Du, X. Jia, Y. Li, and S. C.-H. Huang, “Minimum connected dominating sets and maximal independent sets in unit disk graphs,” Theor. Comp. Sci. 352 (1–3), 1–7 (2006).

    MathSciNet  MATH  Google Scholar 

  139. Y. Wu and Y. Li, “Connecting dominating sets,” in H. Liu, Y.W. Leung, X. Chu (eds), Handbook of Ad Hoc and Sensor Wireless Networks: Architecture, Algorithms and Protocols, pp. 19–39 (2009).

    Google Scholar 

  140. Y. Wu, X. Gao, and Y. Li, “A framework of distributed indexing and data dissemination in large scale wireless sensor networks,” Optim. Lett. 4 (3), 335–345 (2010).

    MathSciNet  MATH  Google Scholar 

  141. L. Wu, H. Du, W. Wu, Y. Hu, A. Wang, and W. Lee, “PTAS for routing-cost constrained minimum connected dominating set in growth bounded graphs,” J. Comb. Optim. 30 (1), 18–26 (2015).

    MathSciNet  MATH  Google Scholar 

  142. M. Yannakakis and F. Gavril, “Edge dominating sets in graphs,” SIAM J. Appl. Math. 38 (3), 364–372 (1980).

    MathSciNet  MATH  Google Scholar 

  143. H.-Y. Yang, C.-H. Lin, and M.-J. Tsai, “Distributed algorithm for efficient construction and maintenance of connected k-hop dominating set in mobile ad hoc networks,” IEEE Trans. Mob. Comput. 7, 444–457 (2008).

    Google Scholar 

  144. J. Y. Yu and P. H. J. Chong, “A survey of clustering schemes for mobile Ad Hoc networks,” IEEE Commun. Surv. & Tut. 7 (1), 32–47 (2005).

    Google Scholar 

  145. R. Yu, X. Wang, and S. K. Das, “EEDTC: energy-efficient dominating tree construction in multi-hop wireless networks,” Pervasive and Mob. Comput. 5 (4), 318–333 (2009).

    Google Scholar 

  146. J. Yu, N. Wang, and G. Wang, “Constructing minimum extended weakly-connected dominating sets for clustering in ad hoc networks,” J. Parallel Distr. Comput. 72 (1), 35–47 (2012).

    MATH  Google Scholar 

  147. J. Yu, N. Wang, G. Wang, and D. Yu, “Connected dominating sets in wireless ad hoc and sensor networks—a comprehensive survey,” Comp. Commun. 36 (2), 121–134 (2013).

    Google Scholar 

  148. Z. Zhang, X. Gao, W. Wu, and D.-Z. Du, “A PTAS for minimum connected dominating set in 3-dimensional wireless sensor networks,” J. Glob. Optim. 45, 451–458 (2009).

    MathSciNet  MATH  Google Scholar 

  149. Z. Zhang, J. Zhou, X. Huang, and D.-Z. Du, “Performance guaranteed approximation algorithm for minimum k-connected m-fold dominating set,” Electr. Prepr., 14 p., Aug. 27, (2016). http://arxiv.org/ abs/1608.07634 [cs.DM].

  150. Y. Zhao, Z. Liao, and L. Miao, “On the algorithmic complexity of edge total domination,” Theor. Comp. Sci. 557, 28–33 (2014).

    MathSciNet  MATH  Google Scholar 

  151. J. Zhou, Z. Zhang, W. Wu, and K. Xing, “A greedy algorithm for the fault-tolerant connected dominating set in a general graph,” J. Comb. Optim. 28 (1), 310–319 (2014).

    MathSciNet  MATH  Google Scholar 

  152. F. Zou, Y. Wang, X.-H. Xu, X. Li, H. Du, P. Wan, and W. Wu, “New approximations for minimum-weighted dominating sets and minimum-weighted connected dominating sets on unit-disk graphs,” Theor. Comp. Sci. 412 (3), 198–208 (2011).

    MathSciNet  MATH  Google Scholar 

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Correspondence to M. Sh. Levin.

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Levin, M.S. Note on Dominating Set Problems. J. Commun. Technol. Electron. 66, S8–S22 (2021). https://doi.org/10.1134/S1064226921130040

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  • Keywords: combinatorial optimization
  • connected dominating sets
  • multicriteria optimization
  • solving schemes
  • networks
  • multiset