Abstract
The total bondage number b t (G) of a graph G with no isolated vertex is the cardinality of a smallest set of edges \({E^{\prime}\subseteq E(G)}\) for which (1) G−E′ has no isolated vertex, and (2) \({\gamma_{t}(G-E^{\prime})>\gamma_{t}(G)}\) . We improve some results on the total bondage number of a graph and give a constructive characterization of a certain class of trees achieving the upper bound on the total bondage number.
References
Bauer D., Harary F., Nieminen J., Suffel C.L.: Domination alteration sets in graphs. Discrete Math. 47, 153–161 (1983)
Domke G.S., Laskar R.C.: The bondage and reinforcement numbers of γ f for some graphs. Discrete Math. 167(168), 249–259 (1997)
Dunbar, J.E., Haynes, T.W., Teschner, U., Volkmann, L.: Bondage, insensitivity and reinforcement. In: T. W. Haynes, S. T. Hedetniemi, P. J. Slater (eds.), Domination in Graphs: Advanced Topics, Marcel Dekker, New York, 471–489 (1998)
Fink J.F., Jacobson M.S., Kinch L.F., Roberts J.: The bondage number of a graph. Discrete Math. 86, 47–57 (1990)
Hartnell B.L., Rall D.F.: Bounds on the bondage number of a graph. Discrete Math. 128, 173–177 (1994)
Haynes, T.W., Hedetniemi, S.T., Slater, P.J.: Fundamentals of Domination in Graphs, Marcel Dekker, New York, (1998)
Henning M.A.: A survey of selected recent results on total domination in graphs. Discrete Math. 309, 32–63 (2009)
Hu, T., Xu, J.-M.: On the Complexity of the Bondage and Reinforcement Problems, Journal of Complexity 28 192–201, (2012). doi:10.1016/j.jco.2011.11.001
Kulli, V.R., Patwari, D.K.: The total bondage number of a graph, Advances in Graph Theory, Vishwa International Publication, 227–235 (1991)
Lu, Y., Xu, J.-M.: The p-Bondage Number of Trees, Graphs and Combinatorics 27:129–141 (2011). doi:10.1007/s00373-010-0956-3
Raczek J.: Paired bondage in trees. Discrete Math. 308, 5570–5575 (2008)
Sridharan N., Elias M. D., Subramanian V. S. A.: Total bondage number of a graph. Akce J. Graphs Combinator. 4, 203–209 (2007)
Teschner U.: New results about the bondage number of a graph. Discrete Math. 171, 249–259 (1997)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.
About this article
Cite this article
Rad, N.J., Raczek, J. Some Progress on Total Bondage in Graphs. Graphs and Combinatorics 30, 717–728 (2014). https://doi.org/10.1007/s00373-013-1303-2
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00373-013-1303-2
Keywords
- Domination number
- Total domination
- Total bondage
- Tree
Mathematics Subject Classification (2000)
- 05C69
- 05C05