Abstract
Given a vertex-coloured graph, a dominating set is said to be tropical if every colour of the graph appears at least once in the set. Here, we study minimum tropical dominating sets from structural and algorithmic points of view. First, we prove that the tropical dominating set problem is NP-complete even when restricted to a simple path. Last, we give approximability and inapproximability results for general and restricted classes of graphs, and establish a FPT algorithm for interval graphs.
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 subscriptionsNotes
- 1.
Also known under the names ‘factor dominating set’ and ‘global dominating set’ in the literature.
References
Akbari, S., Liaghat, V., Nikzad, A.: Colorful paths in vertex-colorings of graphs. Electron. J. Comb. 18, P17 (2011)
Alimonti, P., Kann, V.: Some APX-completeness results for cubic graphs. Theor. Comput. Sci. 237(1–2), 123–134 (2000)
Ausiello, G., Crescenzi, P., Gambosi, G., Kann, V.: Complexity and Approximation. Springer, Heidelberg (1999)
Baker, B.: Approximation algorithms for NP-complete problems on planar graphs. J. ACM 41(1), 153–180 (1994)
Brigham, R.C., Dutton, R.D.: Factor domination in graphs. Discrete Math. 86(1–3), 127–136 (1990)
Caro, Y., Henning, M.A.: Simultaneous domination in graphs. Graphs Combin. 30(6), 1399–1416 (2014)
Chlebík, M., Chlebíková, J.: Approximation hardness of dominating set problems in bounded degree graphs. Inf. Comput. 206(11), 1264–1275 (2008)
Crescenzi, P.: A short guide to approximation preserving reductions. In: Proceedings of the IEEE Conference on Computational Complexity, pp. 262–273 (1997)
Dinur, I., Steurer, D.: Analytical approach to parallel repetition. In: Proceedings of the 46th Annual ACM Symposium on Theory of Computing (STOC 2014), pp. 624–633 (2014)
Duh, R., Fürer, M.: Approximation of k-set cover by semi-local optimization. In: Proceedings of the 29th Annual ACM Symposium on the Theory of Computing (STOC 1997), pp. 256–264 (1997)
Fellows, M., Fertin, G., Hermelin, D., Vialette, S.: Upper and lower bounds for finding connected motifs in vertex-colored graphs. J. Comput. Syst. Sci. 77(4), 799–811 (2011)
Garey, M., Johnson, D.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman, New York (1979)
Haynes, T., Hedetniemi, S., Slater, P.: Fundamentals of Domination in Graphs. Marcel Dekker, New York (1998)
Kann, V.: On the Approximability of NP-complete Optimization Problems. Ph.D. thesis, Department of Numerical Analysis and Computing Science, Royal Institute of Technology, Stockholm (1992)
Li, A.: A generalization of the gallai-roy theorem. Graphs Comb. 17, 681–685 (2001)
Lin, C.: Simple proofs of results on paths representing all colors in proper vertex-colorings. Graphs Comb. 23, 201–203 (2007)
Sampathkumar, E.: The global domination number of a graph. J. Math. Phys. Sci. 23(5), 377–385 (1989)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2016 Springer International Publishing Switzerland
About this paper
Cite this paper
Anglès d’Auriac, JA. et al. (2016). Tropical Dominating Sets in Vertex-Coloured Graphs. In: Kaykobad, M., Petreschi, R. (eds) WALCOM: Algorithms and Computation. WALCOM 2016. Lecture Notes in Computer Science(), vol 9627. Springer, Cham. https://doi.org/10.1007/978-3-319-30139-6_2
Download citation
DOI: https://doi.org/10.1007/978-3-319-30139-6_2
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-30138-9
Online ISBN: 978-3-319-30139-6
eBook Packages: Computer ScienceComputer Science (R0)