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Journal of Combinatorial Optimization

, Volume 22, Issue 2, pp 282–291 | Cite as

On the power domination number of the generalized Petersen graphs

  • Guangjun Xu
  • Liying Kang
Article

Abstract

The problem of monitoring an electric power system by placing as few measurement devices in the system as possible is closely related to the well-known domination problem in graphs. Following a set of rules for power system monitoring, a set S of vertices is defined to be a power dominating set of a graph if every vertex and every edge in the system is monitored by the set S. The minimum cardinality of a power dominating set of G is the power domination number γ p (G). In this paper, we investigate the power domination number for the generalized Petersen graphs, presenting both upper bounds for such graphs and exact results for a subfamily of generalized Petersen graphs.

Keywords

Power domination number The generalized Petersen graph 

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References

  1. Atkins D, Haynes TW, Henning MA (2006) Placing monitoring devices in electric power networks modelled by block graphs. ARS Combin 79:129–143 MathSciNetMATHGoogle Scholar
  2. Behzad A, Behzad M, Praeger CE (2008) On the domination number of the generalized Petersen graphs. Discrete Math 308:603–610 MathSciNetMATHCrossRefGoogle Scholar
  3. Dorbec P, Mollard M, Klavžar S, Špacapan S (2008) Power domination in product graphs. SIAM J Discrete Math 22:554–567 MathSciNetMATHCrossRefGoogle Scholar
  4. Dorfling M, Henning MA (2006) A note on power domination in grid graphs. Discrete Appl Math 154:1023–1027 MathSciNetMATHCrossRefGoogle Scholar
  5. Guo J, Niedermeier R, Raible D (2005) Improved algorithms and complexity results for power domination in graphs. In: Lecture notes comput sci, vol 3623. Springer, Berlin, pp 172–184 Google Scholar
  6. Fox J, Gera R, Stănică P (2007) The independence number for the generalized Petersen graphs. ARS Combin (accepted) Google Scholar
  7. Haynes TW, Hedetniemi SM, Hedetniemi ST, Henning MA (2002) Domination in graphs applied to electric power networks. SIAM J Discrete Math 15:519–529 MathSciNetMATHCrossRefGoogle Scholar
  8. Haynes TW, Hedetniemi ST, Slater PJ (1998) Fundamentals of Domination in Graphs. Marcel Dekker, New York MATHGoogle Scholar
  9. Kneis J, Mölle D, Richter S, Rossmanith P (2006) Parameterized power domination complexity. Inf Process Lett 98:145–149 MATHCrossRefGoogle Scholar
  10. Liao CS, Lee DT (2005) Power domination problem in graphs. In: Lecture notes comput sci, vol 3595. Springer, Berlin, pp 818–828 Google Scholar
  11. Mili L, Baldwin T, Phadke A (1991) Phasor measurement placement for voltage and stability monitoring and control. In: Proceeding of the EPRI-NSF workshop on application of advanced mathematics to power systems. San Franciso, CA Google Scholar
  12. Xu GJ, Kang LY, Shan EF, Zhao M (2006) Power domination in block graphs. Theor Comput Sci 359:299–305 MathSciNetMATHCrossRefGoogle Scholar
  13. Yan H, Kang LY, Xu GJ (2009) The exact domination number of the generalized Petersen graphs. Discrete Math 309:2596–2607 MathSciNetMATHCrossRefGoogle Scholar
  14. Wang HL, Xu XR, Yang YS, Lü K (2009) On the distance paired domination of generalized Petersen graphs P(n,1) and P(n,2). J Comb Optim. doi: 10.1007/s10878-009-9266-1 Google Scholar
  15. Zelinka B (2002) Domination in generalized Petersen graphs. Czechoslovak Math J 52:11–16 MathSciNetMATHCrossRefGoogle Scholar
  16. Zhao M, Kang LY, Chang GJ (2006) Power domination in graphs. Discrete Math 306:1812–1816 MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsThe University of MelbourneParkvilleAustralia
  2. 2.Department of MathematicsShanghai UniversityShanghaiChina

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