Mathematical Notes

, Volume 93, Issue 5–6, pp 795–801 | Cite as

On the bondage number of middle graphs



Let G = (V (G),E(G)) be a simple graph. A subset S of V (G) is a dominating set of G if, for any vertex vV (G) — S, there exists some vertex u ∈ S such that uv ∈ E(G). The domination number, denoted by γ(G), is the cardinality of a minimal dominating set of G. There are several types of domination parameters depending upon the nature of domination and the nature of dominating set. These parameters are bondage, reinforcement, strong-weak domination, strong-weak bondage numbers. In this paper, we first investigate the strong-weak domination number of middle graphs of a graph. Then several results for the bondage, strong-weak bondage number of middle graphs are obtained.


connectivity network design and communication strong and weak domination number bondage number strong and weak bondage number middle graphs 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    E. Sampathkumar and L. Pushpa Latha, “Strong weak domination and domination balance in a graph,” DiscreteMath. 161(1–3), 235–242 (1996).MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    D. Bauer, F. Harary, J. Nieminen, and C. L. Suffel, “Domination alteration sets in graph,” DiscreteMath. 47(No. 2–3), 153–161 (1983).MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    J. F. Fink, M. S. Jacobson, L. F. Kinch, and J. Roberts, “The bondage number of a graph,” DiscreteMath. 86(1–3), 47–57 (1990).MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    D. F. Hartnell and D. F. Rall, “Bounds on the bondage number of a graph,” DiscreteMath. 128(1-3), 173–177 (1994).MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    J. Ghoshal, R. Laskar, D. Pillone, and C. Wallis, “Strong bondage and strong reinforcement numbers of graphs,” Congr. Numer. 108, 33–42 (1995).MathSciNetMATHGoogle Scholar
  6. 6.
    K. Ebadi and L. Pushpalatha, “Smarandachely bondage number of a graph,” Int. J. Math. Comb. 4, 09–19 (2010).MathSciNetGoogle Scholar
  7. 7.
    M. Nihei, “Algebraic connectivity of the line graph, the middle graph and the total graph of a regular graph,” Ars Combin. 69, 215–221 (2003).MathSciNetMATHGoogle Scholar
  8. 8.
    A. Mamut and E. Vumar, “A note on the integrity of middle graphs,” in Discrete Geometry, Combinatorics, and Graph Theory, Lecture Notes in Comput. Sci. (Springer-Verlag, Berlin, 2007), Vol. 4381, pp. 130–134.Google Scholar

Copyright information

© Pleiades Publishing, Ltd. 2013

Authors and Affiliations

  1. 1.Ege UniversityEgeTurkey
  2. 2.Izmir University of EconomicsIzmirTurkey

Personalised recommendations