Journal of Combinatorial Optimization

, Volume 27, Issue 2, pp 241–255 | Cite as

Signed Roman domination in graphs

  • H. Abdollahzadeh Ahangar
  • Michael A. Henning
  • Christian Löwenstein
  • Yancai Zhao
  • Vladimir Samodivkin


In this paper we continue the study of Roman dominating functions in graphs. A signed Roman dominating function (SRDF) on a graph G=(V,E) is a function f:V→{−1,1,2} satisfying the conditions that (i) the sum of its function values over any closed neighborhood is at least one and (ii) for every vertex u for which f(u)=−1 is adjacent to at least one vertex v for which f(v)=2. The weight of a SRDF is the sum of its function values over all vertices. The signed Roman domination number of G is the minimum weight of a SRDF in G. We present various lower and upper bounds on the signed Roman domination number of a graph. Let G be a graph of order n and size m with no isolated vertex. We show that \(\gamma _{\mathrm{sR}}(G) \ge\frac{3}{\sqrt{2}} \sqrt{n} - n\) and that γ sR(G)≥(3n−4m)/2. In both cases, we characterize the graphs achieving equality in these bounds. If G is a bipartite graph of order n, then we show that \(\gamma_{\mathrm{sR}}(G) \ge3\sqrt{n+1} - n - 3\), and we characterize the extremal graphs.


Roman domination Signed domination Signed Roman domination 



The authors thank the referees for their helpful comments and suggestions to improve the exposition and readability of the paper.

Research of the second author was supported in part by the South African National Research Foundation and the University of Johannesburg. Research of the fourth author was partially supported by the Nature Science Foundation of Anhui Provincial Education Department (No. KJ2011B090). Research of the third author was supported by the Deutsche Forschungsgemeinschaft (GZ: LO 1758/1-1).


  1. Chambers W, Kinnersley B, Prince N, West DB (2009) Extremal problems for Roman domination. SIAM J Discrete Math 23:1575–1586 CrossRefMATHMathSciNetGoogle Scholar
  2. Cockayne EJ, Dreyer Jr PA, Hedetniemi SM, Hedetniemi ST (2004) Roman domination in graphs. Discrete Math 278:11–22 CrossRefMATHMathSciNetGoogle Scholar
  3. Cockayne EJ, Grobler PJP, Grundlingh W, Munganga J, van Vuuren JH (2005) Protection of a graph. Util Math 67:19–32 MATHMathSciNetGoogle Scholar
  4. Dunbar J, Hedetniemi ST, Henning MA, Slater PJ (1995) Signed domination in graphs. Graph Theory Comb Appl 1:311–322 MathSciNetGoogle Scholar
  5. Favaron O (1996) Signed domination in regular graphs. Discrete Math 158:287–293 CrossRefMATHMathSciNetGoogle Scholar
  6. Fernau H (2008) Roman domination: a parameterized perspective. Int J Comput Math 85:25–38 CrossRefMATHMathSciNetGoogle Scholar
  7. Furedi Z, Mubayi D (1999) Signed domination in regular graphs and set-systems. J Comb Theory, Ser B 76:223–239 CrossRefMathSciNetGoogle Scholar
  8. Haynes TW, Hedetniemi ST, Slater PJ (1998) Fundamentals of domination in graphs. Dekker, New York MATHGoogle Scholar
  9. Henning MA (1998) Dominating functions in graphs. In: Haynes TW, Hedetniemi ST, Slater PJ (eds) Domination in graphs: vol II. Dekker, New York, pp 31–62 Google Scholar
  10. Henning MA (2002) A characterization of Roman trees. Discuss Math, Graph Theory 22(2):325–334 CrossRefMATHMathSciNetGoogle Scholar
  11. Henning MA (2003) Defending the Roman empire from multiple attacks. Discrete Math 271:101–115 CrossRefMATHMathSciNetGoogle Scholar
  12. Henning MA, Hedetniemi ST (2003) Defending the Roman empire–a new strategy. Discrete Math 266:239–251 CrossRefMATHMathSciNetGoogle Scholar
  13. ReVelle CS (1997) Can you protect the Roman empire? Johns Hopkins Mag 49(2):40 Google Scholar
  14. ReVelle CS, Rosing KE (2000) Defendens imperium Romanum: a classical problem in military strategy. Am Math Mon 107(7):585–594 CrossRefMATHMathSciNetGoogle Scholar
  15. Song X, Wang X (2006) Roman domination number and domination number of a tree. Chin Q J Math 21:358–367 MathSciNetGoogle Scholar
  16. Stewart I (1999) Defend the Roman empire! Sci Am December:136–138 CrossRefGoogle Scholar
  17. Zhang Z, Xu B, Li Y, Liu L (1999) A note on the lower bounds of signed domination number of a graph. Discrete Math 195:295–298 CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  • H. Abdollahzadeh Ahangar
    • 1
  • Michael A. Henning
    • 2
  • Christian Löwenstein
    • 2
  • Yancai Zhao
    • 3
  • Vladimir Samodivkin
    • 4
  1. 1.Babol University of TechnologyBabolIran
  2. 2.University of JohannesburgJohannesburgSouth Africa
  3. 3.Wuxi City College of Vocational TechnologyJiangsuChina
  4. 4.University of Architecture Civil Engineering and GeodesySofiaBulgaria

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