Abstract
A total vertex cover is a vertex cover whose induced subgraph consists of a set of connected components, each of which contains at least two vertices. A t-total vertex cover is a total vertex cover where each component of its induced subgraph contains at least t vertices. The total vertex cover (TVC) problem and the t-total vertex cover (t-TVC) problem ask for the corresponding cover set with minimum cardinality, respectively. In this paper, we first show that the t-TVC problem is NP-complete for connected subcubic grid graphs of arbitrarily large girth. Next, we show that the t-TVC problem is NP-complete for 3-connected cubic planar graphs. Moreover, we show that the t-TVC problem is APX-complete for connected subcubic graphs of arbitrarily large girth.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsReferences
Alimonti, P., Kann, V.: Some APX-completeness results for cubic graphs. Theoret. Comput. Sci. 237(1–2), 123–134 (2000)
Crescenzi, P.: A short guide to approximation preserving reductions. In: Proceedings of the 12th Annual IEEE Conference on Computational Complexity, pp. 262–273 (1997)
de Berg, M., Khosravi, A.: Optimal binary space partitions in the plane. In: Proceedings of the 16th International Conference on Computing and Combinatorics, pp. 329–343 (2010)
Fernau, H., Manlove, D.F.: Vertex and edge covers with clustering properties: complexity and algorithms. J. Discrete Algorithms 7(2), 149–167 (2009)
Garey, M.R., Johnson, D.S.: The rectilinear steiner tree problem is NP-complete. SIAM J. Appl. Math. 32(4), 826–834 (1977)
Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman & Co., New York (1990)
Gross, J.L., Yellen, J.: Graph Theory and Its Applications. CRC Press, Boca Raton (2005)
Hopcroft, J.E., Karp, R.M.: A \(n^{5/2}\) algorithm for maximum matchings in bipartite. In: Proceedings of 12th Annual IEEE Symposium on Switching and Automata Theory, pp. 225–231 (1971)
Karp, R.M.: Reducibility among combinatorial problems. In: Proceedings of a Symposium on the Complexity of Computer Computations, pp. 85–103 (1972)
König, D.: Vgráfok és mátrixok. Matematikai és Fizikai Lapok 38, 116–119 (1931)
Murphy, O.J.: Computing independent sets in graphs with large girth. Discrete Appl. Math. 35(2), 167–170 (1992)
Uehara, R.: NP-complete problems on a 3-connected cubic planar graph and their applications. Technical Report TWCU-M-0004, Tokyo Woman’s Christian University (1996)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2017 Springer International Publishing AG
About this paper
Cite this paper
Poon, SH., Wang, WL. (2017). On Complexity of Total Vertex Cover on Subcubic Graphs. In: Gopal, T., Jäger , G., Steila, S. (eds) Theory and Applications of Models of Computation. TAMC 2017. Lecture Notes in Computer Science(), vol 10185. Springer, Cham. https://doi.org/10.1007/978-3-319-55911-7_37
Download citation
DOI: https://doi.org/10.1007/978-3-319-55911-7_37
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-55910-0
Online ISBN: 978-3-319-55911-7
eBook Packages: Computer ScienceComputer Science (R0)