Abstract
Let G = (V,E) be a simple graph with vertex set V and edge set E. A subset W ⊆ V ∪ E is a total covering set if every element x ∈ (V ∪ E) ∖ W is either adjacent to or incident to an element of W. The total covering problem is to find a total covering set of G. In this paper, we show that this problem can be solved in linear-time on block-cactus graphs.
This work was supported in part by the National Science Council of the Republic of China under contracts NSC 100–2221–E–011–067-MY3 and NSC 101–2221–E–011–038–MY3.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Alavi, Y., Behzad, M., Lesniak-Foster, L.M., Nordhaus, E.A.: Total matchings and total coverings of graphs. J. Graph Theor. 1, 135–140 (1977)
Alavi, Y., Liu, J., Wang, J., Zhang, Z.: On total covers of graphs. Discrete Math. 100, 229–233 (1992)
Cockayne, E.J., Goodman, S., Hedetniemi, S.T.: A linear algorithm for the domination number of a tree. Inform. Process. Lett. 4, 41–44 (1975)
Erdős, P., Meir, A.: On total matching numbers and total covering numbers of complementary graphs. Discrete Math. 19, 229–233 (1977)
Haynes, T.W., Hedetniemi, S.T., Slater, P.J.: Fundamentals of Domination in Graphs. Marcel Dekker, New York (1998)
Meir, A.: On total covering and matching of graphs. J. Comb. Theory B 24, 164–168 (1978)
Mili, L., Baldwin, T., Phadke, A.: Phasor measurement placement for voltage and stability monitoring and control. In: Proc. of the EPRI-NSF Workshop on Application of Advanced Mathematics to Power System, San Francisco, CA (1991)
Natarajan, K.S., White, L.J.: Optimum domination in weighted trees. Inform. Process. Lett. 7, 261–265 (1978)
Rautenbach, D., Volkmann, L.: The domatic number of block-cactus graphs. Discrete Math. 187, 185–193 (1998)
Zhao, Y.C., Kang, L.Y., Sohn, M.Y.: The algorithmic complexity of mixed domination in graphs. Theore. Comput. Sci. 412, 2387–2392 (2011)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Li, YT., Liu, JJ., Wang, YL. (2013). On Total Covers of Block-Cactus Graphs. In: Chang, RS., Jain, L., Peng, SL. (eds) Advances in Intelligent Systems and Applications - Volume 1. Smart Innovation, Systems and Technologies, vol 20. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-35452-6_5
Download citation
DOI: https://doi.org/10.1007/978-3-642-35452-6_5
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-35451-9
Online ISBN: 978-3-642-35452-6
eBook Packages: EngineeringEngineering (R0)