Abstract
We study the parameterized complexity of the pseudo-achromatic number problem: Given an undirected graph and a parameter k, determine if the graph can be partitioned into k groups such that every two groups are connected by at least one edge. This problem has been extensively studied in graph theory and combinatorial optimization. We show that the problem has a kernel of at most (k − 2)(k + 1) vertices that is constructable in time \(O(m\sqrt{n})\), where n and m are the number of vertices and edges, respectively, in the graph, and k is the parameter. This directly implies that the problem is fixed-parameter tractable. We also study generalizations of the problem and show that they are parameterized intractable.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Balasubramanian, R., Raman, V., Yegnanarayanan, V.: On the pseudoachromatic number of join of graphs. International Journal of Computer Mathematics 80(9), 1131–1137 (2003)
Bhave, V.: On the pseudoachromatic number of a graph. Fundamenta Mathematicae 102(3), 159–164 (1979)
Bodlaender, H.: Achromatic number is NP-complete for cographs and interval graphs. Information Processing Letters 32(3), 135–138 (1989)
Bollobás, B., Reed, B., Thomason, A.: An extremal function for the achromatic number. Graph Structure Theory, 161–166 (1991); pp. 18–37 (2005)
Chen, J., Huang, X., Kanj, I., Xia, G.: Strong computational lower bounds via parameterized complexity. Journal of Computer and System Sciences 72(8), 1346–1367 (2006)
Chen, J., Meng, J., Kanj, I., Xia, G., Zhang, F.: On the pseudo-achromatic number problem. Technical report 08-006 at: http://www.cdm.depaul.edu/research/Pages/TechnicalReports.aspx.
Cormen, T., Leiserson, C., Rivest, R., Stein, C.: Introduction to Algorithms, 2nd edn. McGraw-Hill Book Company, Boston (2001)
Downey, R., Fellows, M.: Parameterized Complexity. Springer, Heidelberg (1999)
Edwards, K., McDiarmid, C.: The complexity of harmonious coloring for trees. Discrete Applied Mathematics 57, 133–144 (1995)
Downey, R., Fellows, M., Stege, U.: Parameterized complexity: a framework for systematically confronting computational intractability, in Contemporary Trends in Discrete Mathematics. In: Graham, R., Kratochvíl, J., Nešetřil, J., Roberts, F. (eds.) Proc. DIMACS-DIMATIA Workshop, Prague. AMS-DIMACS Series in Discrete Mathematics and Theoretical Computer Science, vol. 49, pp. 49–99 (1997)
Gupta, R.P.: Bounds on the chromatic and achromatic numbers of complementary graphs. In: Tutte, W.T. (ed.) Recnet Progress in Combinatorics, Proc. 3rd Waterloo Conference on Combinatorics, Waterloo, pp. 229–235. Academic Press, New York (1969)
Kortsarz, G., Radhakrishnan, J., Sivasubramanian, S.: Complete partitions of graphs. In: Proc. 16th Annual ACM-SIAM symposium on Discrete algorithms, pp. 860–869 (2005)
Sampathkumar, E., Bhave, V.: Partition graphs and coloring numbers of graphs. Discrete Mathematics 16, 57–60 (1976)
Yegnanarayanan, V.: On pseudocoloring of graphs. Utilitas Mathematica 62, 199–216 (2002)
Yegnanarayanan, V.: The pseudoachromatic number of a graph. Southern Aian Bulletin of Mathematics 24, 129–136 (2002)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2008 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Chen, J., Kanj, I.A., Meng, J., Xia, G., Zhang, F. (2008). On the Pseudo-achromatic Number Problem. In: Broersma, H., Erlebach, T., Friedetzky, T., Paulusma, D. (eds) Graph-Theoretic Concepts in Computer Science. WG 2008. Lecture Notes in Computer Science, vol 5344. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-92248-3_8
Download citation
DOI: https://doi.org/10.1007/978-3-540-92248-3_8
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-92247-6
Online ISBN: 978-3-540-92248-3
eBook Packages: Computer ScienceComputer Science (R0)