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Approximation algorithm for a generalized Roman domination problem in unit ball graphs

  • Limin Wang
  • Yalin Shi
  • Zhao Zhang
  • Zan-Bo Zhang
  • Xiaoyan ZhangEmail author
Article
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Abstract

In this paper we propose a generalized Roman domination problem called connected strong k-Roman dominating set problem. It is NP-hard even in a unit ball graph. Unit ball graphs are the intersection graphs of equal sized balls in the three-dimensional space, they are widely used as a mathematical model for wireless sensor networks and some problems in computational geometry. This paper presents the first constant approximation algorithm with a guaranteed performance ratio at most \(6(k+2)\) in unit ball graphs, where k is a positive integer.

Keywords

Connected strong k-Roman dominating set Constant approximation algorithm Unit ball graph 
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Notes

Acknowledgements

The authors would like to thank the anonymous reviewers for their valuable suggestions and comments that have helped a lot to improve the quality of the paper.

References

  1. Akyildiz IF, Pompili D, Melodia T (2005) Underwater acoustic sensor networks: research challenges. Ad Hoc Netw 3:257–279CrossRefGoogle Scholar
  2. Cockayne EJ, Dreyer PA Jr, Hedetniemi SM, Hedetniemi ST (2004) Roman domination in graphs. Discrete Math 278(1–3):11–22MathSciNetCrossRefGoogle Scholar
  3. Chambers EW, Kinnersley B, Prince N, West DB (2009) Extremal problem for Roman domination. SIAM J Discrete Math 23(3):1575–1586MathSciNetCrossRefGoogle Scholar
  4. Dreyer PA Jr (2000) Applications and variations of domination in graphs. Rutgers University, New Jersey PhD ThesisGoogle Scholar
  5. Henning MA (2003) Defending the roman empire from multiple attacks. Discrete Math 271(1–3):101–115MathSciNetCrossRefGoogle Scholar
  6. Henning MA, Hedetniemi ST (2003) Defending the Roman empire—a new strategy. Discrete Math 266:239–251MathSciNetCrossRefGoogle Scholar
  7. Huang H, Richa AW, Segal M (2004) Approximation algorithms for the mobile piercing set problem with applications to clustering in ad hoc networks. Mob Netw Appl 9(2):151–161CrossRefGoogle Scholar
  8. Lideloff M, Kloks T, Liu J, Peng SL (2005) Roman domination over some graph classes. In: Kratsch D (ed) WG 2005. LNCS 3787, pp 103–114Google Scholar
  9. Liu CH, Chang GJ (2012) Roman domination on 2-connected graphs. SIAM J Discrete Math 26(1):193–205MathSciNetCrossRefGoogle Scholar
  10. Pagourtzis A, Penna P, Schlude K, Steinhofel K, Taylor DS, Widmayer P (2002) Server placements, Roman domination and other dominating set variants. Found Inf Technol Era Netw Mob Comput 96:280–291CrossRefGoogle Scholar
  11. Shang WP, Hu XD (2007) The Roman domination problem in unit disk graphs. In: Shi Y et al (eds) ICCS 2007. Part III, LNCS 4489, pp 305–312Google Scholar
  12. Shang WP, Wang XM, Hu XD (2010) Roman domination and the variants in unit disk graphs. Discrete Math Algorithms Appl 2(1):99–105MathSciNetCrossRefGoogle Scholar
  13. Stewart I (1999) Defend the Roman empire!. Sci Am 281(6):136–138CrossRefGoogle Scholar
  14. Zhu EQ, Shao ZH (2018) Extremal problems on weak Roman domination number. Inf Process Lett 138:12–18MathSciNetCrossRefGoogle Scholar

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

Authors and Affiliations

  1. 1.State Key Laboratory for Novel Software TechnologyNanjing UniversityNanjingChina
  2. 2.School of Mathematical Science and Institute of MathematicsNanjing Normal UniversityNanjingChina
  3. 3.College of Mathematics and Computer ScienceZhejiang Normal UniversityJinhuaChina
  4. 4.Institute of Artificial Intelligence and Deep Learning, and School of Statistics and MathematicsGuangdong University of Finance & EconomicsGuangzhouChina

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