Abstract
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 Ʃe∈E(C)f(e) ≥ 1 for every induced cycle C of G. The signed cycle domination number of G is defined as γ′sc(G) = min{Ʃe∈Ef(e)| f is an SCDF of G}. This paper will characterize all maximal planar graphs G with order n ≥ 6 and γ′sc(G) = n.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bondy, J. A., Murty, V. S. R.: Graph Theory with Applications, Elsevier, Amsterdam, 1976
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)
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)
Pi, X., Liu, H.: On the signed edge domination numbers of K m,n. Ars Combin., 112, 471–478 (2013)
Pi, X.: On the characterization of graphs with given signed cycle domination number. Adv. Math. China, 2(44), 219–228 (2015)
Xu, B.: On signed cycle domination in graphs. Discrete Math., 309, 1007–1012 (2009)
Xu, B.: On signed edge domination numbers of graphs. Discrete Math., 239, 179–189 (2001)
Author information
Authors and Affiliations
Corresponding author
Additional information
Supported by Doctoral Scientific Research Fund of Harbin Normal University (Grant No. KGB201008)
Rights and permissions
About this article
Cite this article
Pi, X.M. On the Characterization of Maximal Planar Graphs with a Given Signed Cycle Domination Number. Acta. Math. Sin.-English Ser. 34, 911–920 (2018). https://doi.org/10.1007/s10114-017-6283-3
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10114-017-6283-3
Keywords
- Domination number
- signed cycle domination function
- signed cycle domination number
- planar graph
- maximal planar graph