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On the Roman Bondage Number of Planar Graphs

Graphs and Combinatorics volume 27, pages 531–538 (2011)Cite this article

Abstract

A Roman dominating function on a graph G is a function f : V(G) → {0, 1, 2} satisfying the condition that every vertex u for which f (u) = 0 is adjacent to at least one vertex v for which f (v) = 2. The weight of a Roman dominating function is the value \({f (V(G)) = \sum_{u\in V(G)} f (u)}\). The Roman domination number, γ R (G), of G is the minimum weight of a Roman dominating function on G. The Roman bondage number b R (G) of a graph G with maximum degree at least two is the minimum cardinality of all sets \({E^{\prime} \subseteq E(G)}\) for which γ R (G − E′) > γ R (G). In this paper we present different bounds on the Roman bondage number of planar graphs.

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References

  1. Bauer D., Harary F., Nieminen J., Suffel C.L.: Domination alteration sets in graphs. Discrete Math. 47, 153–161 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  2. Carlson K., Develin M.: On the bondage number of planar and directed graphs. Discrete Math. 306, 820–826 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  3. Cockayne E.J., Dreyer P.M. Jr, Hedetniemi S.M., Hedetniemi S.T.: Roman domination in graphs. Discrete Math. 278, 11–22 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  4. Dunbar J.E., Haynes T.W., Teschner U., Volkmann L.: Bondage, insensitivity, and reinforcement. In: Haynes, T.W., Hedetniemi, S.T., Slater, P.J. (eds) Domination in Graphs: Advanced Topics, pp. 471–489. Marcel Dekker, New York (1998)

    Google Scholar 

  5. Fink J.F., Jacobson M.S., Kinch L.F., Roberts J.: The bondage number of a graph. Discrete Math. 86, 47–57 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  6. Fischermann M., Rautenbach D., Volkmann L.: Remarks on the bondage number of planar grahs. Discrete Math. 260, 57–67 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  7. Hartnell B.L., Rall D.F.: Bounds on the bondage number of a graph. Discrete Math. 128, 173–177 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  8. Haynes T.W., Hedetniemi S.T., Slater P.J.: Fundamentals of Domination in Graphs. Marcel Dekker, New York (1998)

    MATH  Google Scholar 

  9. Jafari Rad, N., Volkmann, L.: Roman bondage in graphs. Submitted for publication (2009)

  10. Kang L., Yuan J.: Bondage number of planar graphs. Discrete Math. 222, 191–198 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  11. ReVelle C.S., Rosing K.E.: Defendens imperium romanum: a classical problem in military strategy. Am. Math. Mon. 107, 585–594 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  12. Stewart I.: Defend the Roman Empire!. Sci. Am. 281, 136–139 (1999)

    Article  Google Scholar 

  13. Teschner U.: New results about the bondage number of a graph. Discrete Math. 171, 249–259 (1997)

    Article  MATH  MathSciNet  Google Scholar 

Download references

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Authors and Affiliations

  1. Department of Mathematics, Shahrood University of Technology, Shahrood, Iran

    Nader Jafari Rad

  2. School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P.O. Box 19395-5746, Tehran, Iran

    Nader Jafari Rad

  3. Lehrstuhl II für Mathematik, RWTH Aachen University, Templergraben 55, 52056, Aachen, Germany

    Lutz Volkmann

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  1. Nader Jafari Rad
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  2. Lutz Volkmann
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Correspondence to Nader Jafari Rad.

Additional information

The research of first author was in part supported by a grant from IPM (No. 89050040).

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Jafari Rad, N., Volkmann, L. On the Roman Bondage Number of Planar Graphs. Graphs and Combinatorics 27, 531–538 (2011). https://doi.org/10.1007/s00373-010-0978-x

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  • DOI: https://doi.org/10.1007/s00373-010-0978-x

  • Domination
  • Roman domination
  • Roman bondage number
  • Planar graphs