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On the power domination number of the generalized Petersen graphs

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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.

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References

  • Atkins D, Haynes TW, Henning MA (2006) Placing monitoring devices in electric power networks modelled by block graphs. ARS Combin 79:129–143

    MathSciNet  MATH  Google Scholar 

  • Behzad A, Behzad M, Praeger CE (2008) On the domination number of the generalized Petersen graphs. Discrete Math 308:603–610

    Article  MathSciNet  MATH  Google Scholar 

  • Dorbec P, Mollard M, Klavžar S, Špacapan S (2008) Power domination in product graphs. SIAM J Discrete Math 22:554–567

    Article  MathSciNet  MATH  Google Scholar 

  • Dorfling M, Henning MA (2006) A note on power domination in grid graphs. Discrete Appl Math 154:1023–1027

    Article  MathSciNet  MATH  Google Scholar 

  • 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 

  • Fox J, Gera R, Stănică P (2007) The independence number for the generalized Petersen graphs. ARS Combin (accepted)

  • Haynes TW, Hedetniemi SM, Hedetniemi ST, Henning MA (2002) Domination in graphs applied to electric power networks. SIAM J Discrete Math 15:519–529

    Article  MathSciNet  MATH  Google Scholar 

  • Haynes TW, Hedetniemi ST, Slater PJ (1998) Fundamentals of Domination in Graphs. Marcel Dekker, New York

    MATH  Google Scholar 

  • Kneis J, Mölle D, Richter S, Rossmanith P (2006) Parameterized power domination complexity. Inf Process Lett 98:145–149

    Article  MATH  Google Scholar 

  • Liao CS, Lee DT (2005) Power domination problem in graphs. In: Lecture notes comput sci, vol 3595. Springer, Berlin, pp 818–828

    Google Scholar 

  • 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

  • Xu GJ, Kang LY, Shan EF, Zhao M (2006) Power domination in block graphs. Theor Comput Sci 359:299–305

    Article  MathSciNet  MATH  Google Scholar 

  • Yan H, Kang LY, Xu GJ (2009) The exact domination number of the generalized Petersen graphs. Discrete Math 309:2596–2607

    Article  MathSciNet  MATH  Google Scholar 

  • 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 

  • Zelinka B (2002) Domination in generalized Petersen graphs. Czechoslovak Math J 52:11–16

    Article  MathSciNet  MATH  Google Scholar 

  • Zhao M, Kang LY, Chang GJ (2006) Power domination in graphs. Discrete Math 306:1812–1816

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics and Statistics, The University of Melbourne, Parkville, VIC, 3010, Australia

    Guangjun Xu

  2. Department of Mathematics, Shanghai University, Shanghai, 200444, China

    Liying Kang

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  1. Guangjun Xu
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  2. Liying Kang
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Correspondence to Guangjun Xu.

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Xu, G., Kang, L. On the power domination number of the generalized Petersen graphs. J Comb Optim 22, 282–291 (2011). https://doi.org/10.1007/s10878-010-9293-y

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