Acta Mathematica Sinica, English Series

, Volume 34, Issue 5, pp 911–920 | Cite as

On the Characterization of Maximal Planar Graphs with a Given Signed Cycle Domination Number



Let G = (V, E) be a simple graph. A function f : E → {+1,−1} is called a signed cycle domination function (SCDF) of G if ƩeE(C)f(e) ≥ 1 for every induced cycle C of G. The signed cycle domination number of G is defined as γsc(G) = min{ƩeEf(e)| f is an SCDF of G}. This paper will characterize all maximal planar graphs G with order n ≥ 6 and γsc(G) = n.


Domination number signed cycle domination function signed cycle domination number planar graph maximal planar graph 

MR(2010) Subject Classification



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  1. [1]
    Bondy, J. A., Murty, V. S. R.: Graph Theory with Applications, Elsevier, Amsterdam, 1976CrossRefMATHGoogle Scholar
  2. [2]
    Guan, J., Liu, X., Lu, C., et al.: Three conjectures on the signed cycle domination in graphs. J. Comb Optim., 4(25), 639–645 (2013)MathSciNetCrossRefMATHGoogle Scholar
  3. [3]
    Pi, X., Liu, H.: On the characterization of trees with signed edge domination numbers 1, 2, 3, or 4. Discrete Math., 309, 1779–1782 (2009)MathSciNetCrossRefMATHGoogle Scholar
  4. [4]
    Pi, X., Liu, H.: On the signed edge domination numbers of K m,n. Ars Combin., 112, 471–478 (2013)MathSciNetMATHGoogle Scholar
  5. [5]
    Pi, X.: On the characterization of graphs with given signed cycle domination number. Adv. Math. China, 2(44), 219–228 (2015)MathSciNetMATHGoogle Scholar
  6. [6]
    Xu, B.: On signed cycle domination in graphs. Discrete Math., 309, 1007–1012 (2009)MathSciNetCrossRefMATHGoogle Scholar
  7. [7]
    Xu, B.: On signed edge domination numbers of graphs. Discrete Math., 239, 179–189 (2001)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Chinese Mathematical Society and Springer-Verlag GmbH Germany, part of Springer Nature 2017

Authors and Affiliations

  1. 1.Department of MathematicsHarbin Normal UniversityHarbinP. R. China

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