Abstract
A graph G = (V,E) is a 3-leaf power iff there exists a tree T the leaf set of which is V and such that (u,v) ∈ E iff u and v are at distance at most 3 in T. The 3-leaf power edge modification problems, i.e. edition (also known as the Closest 3-Leaf Power), completion and edge-deletion are FPT when parameterized by the size of the edge set modification. However, a polynomial kernel was known for none of these three problems. For each of them, we provide a kernel with O(k 3) vertices that can be computed in linear time. We thereby answer an open question first mentioned by Dom, Guo, Hüffner and Niedermeier [9].
Work supported by the French research grant ANR-06-BLAN-0148-01 ”Graph Decomposition and Algorithms - GRAAL”.
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
Barthélémy, J.-P., Guénoche, A.: Trees and proximity representations. John Wiley & Sons, Chichester (1991)
Böcker, S., Briesemeister, S., Klau, G.W.: Exact algorithms for cluster editing: evaluation and experiments. In: McGeoch, C.C. (ed.) WEA 2008. LNCS, vol. 5038, pp. 289–302. Springer, Heidelberg (2008)
Brandstädt, A., Le., V.B.: Structure and linear time recognition of 4-leaf powers. ACM Transactions on Algorithms (to appear)
Brandstädt, A., Le, V.B.: Structure and linear time recognition of 3-leaf powers. Information Processing Letters 98(4), 133–138 (2006)
Chang, M.-S., Ko, M.-T.: The 3-steiner root problem. In: Brandstädt, A., Kratsch, D., Müller, H. (eds.) WG 2007. LNCS, vol. 4769, pp. 109–120. Springer, Heidelberg (2007)
Dehne, F., Langston, M., Luo, X., Pitre, S., Shaw, P., Zhang, Y.: The cluster editing problem: implementations and experiments. In: Bodlaender, H.L., Langston, M.A. (eds.) IWPEC 2006. LNCS, vol. 4169, pp. 13–24. Springer, Heidelberg (2006)
Dom, M., Guo, J., Hüffner, F., Neidermeier, R.: Error compensation in leaf root problems. Algorithmica 44, 363–381 (2006)
Dom, M., Guo, J., Hüffner, F., Niedermeier, R.: Error compensation in leaf root problems. In: Fleischer, R., Trippen, G. (eds.) ISAAC 2004. LNCS, vol. 3341, pp. 389–401. Springer, Heidelberg (2004)
Dom, M., Guo, J., Hüffner, F., Niedermeier, R.: Extending the tractability border for closest leaf powers. In: Kratsch, D. (ed.) WG 2005. LNCS, vol. 3787, pp. 397–408. Springer, Heidelberg (2005)
Dom, M., Guo, J., Hüffner, F., Niedermeier, R.: Closest 4-leaf power is fixed-parameter tractable. Discrete Applied Mathematics 156(18), 3345–3361 (2008)
Downey, R.G., Fellows, M.R.: Parameterized complexity. Springer, Heidelberg (1999)
Fellows, M.R., Langston, M., Rosamond, F., Shaw, P.: Efficient parameterized preprocessing for cluster editing. In: Csuhaj-Varjú, E., Ésik, Z. (eds.) FCT 2007. LNCS, vol. 4639, pp. 312–321. Springer, Heidelberg (2007)
Goldberg, P.W., Golumbic, M.C., Kaplan, H., Shamir, R.: Four strikes against physical mapping of DNA. Journal of Computational Biology 2, 139–152 (1995)
Gramm, J., Guo, J., Hüffner, F., Niedermeier, R.: Graph-modeled data clustering: Exact algorithms for clique generation. Theory of Computing Systems 38(4), 373–392 (2005)
Guo, J.: A more effective linear kernelization for cluster editing. In: Chen, B., Paterson, M., Zhang, G. (eds.) ESCAPE 2007. LNCS, vol. 4614, pp. 36–47. Springer, Heidelberg (2007)
Guo, J., Niedermeier, R.: Invitation to data reduction and problem kernelization. SIGACT News 38(1), 31–45 (2007)
Hansen, P., Jaumard, B.: Cluster analysis and mathematical programming. Math. Program. 79(1-3), 191–215 (1997)
Kaplan, H., Shamir, R., Tarjan, R.E.: Tractability of parameterized completion problems on chordal, strongly chordal, and proper interval graphs. SIAM J. Comput 28, 780–791 (1999)
Kearney, P., Corneil, D.: Tree powers. Journal of Algorithms 29(1), 111–131 (1998)
Lin, G.H., Kearney, P.E., Jiang, T.: Phylogenetic k-root and steiner k-root. In: Lee, D.T., Teng, S.-H. (eds.) ISAAC 2000. LNCS, vol. 1969, pp. 539–551. Springer, Heidelberg (2000)
Natazon, A., Shamir, R., Sharan, R.: Complexity classification of some edge modification problems. Discrete Applied Mathematics 113, 109–128 (2001)
Neidermeier, R.: Invitation to fixed parameter algorithms. Oxford Lectures Series in Mathematics and its Applications, vol. 31. Oxford University Press, Oxford (2006)
Nishimura, N., Ragde, P., Thilikos, D.: On graph powers for leaf-labeled trees. Journal of Algorithms 42(1), 69–108 (2002)
Protti, F., Dantas da Silva, M., Szwarcfiter, J.L.: Applying modular decomposition to parameterized cluster editing problems. Theory of Computing Systems (2007)
Shamir, R., Sharan, R., Tsur, D.: Cluster graph modification problems. Discrete Applied Mathematics 144(1-2), 173–182 (2004)
Tedder, M., Corneil, D., Habib, M., Paul, C.: Simpler linear-time modular decomposition via recursive factorizing permutations. In: Aceto, L., Damgård, I., Goldberg, L.A., Halldórsson, M.M., Ingólfsdóttir, A., Walukiewicz, I. (eds.) ICALP 2008, Part I. LNCS, vol. 5125, pp. 634–645. Springer, Heidelberg (2008)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2009 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Bessy, S., Paul, C., Perez, A. (2009). Polynomial Kernels for 3-Leaf Power Graph Modification Problems. In: Fiala, J., Kratochvíl, J., Miller, M. (eds) Combinatorial Algorithms. IWOCA 2009. Lecture Notes in Computer Science, vol 5874. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-10217-2_10
Download citation
DOI: https://doi.org/10.1007/978-3-642-10217-2_10
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-10216-5
Online ISBN: 978-3-642-10217-2
eBook Packages: Computer ScienceComputer Science (R0)