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A Polynomial Kernel for Block Graph Deletion

Authors Eun Jung Kim, O-joung Kwon

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LIPIcs.IPEC.2015.270.pdf
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  • block graph
  • polynomial kernel
  • single-exponential FPT algorithm

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Abstract

In the Block Graph Deletion problem, we are given a graph G on n vertices and a positive integer k, and the objective is to check whether it is possible to delete at most k vertices from G to make it a block graph, i.e., a graph in which each block is a clique. In this paper, we obtain a kernel with O(k^{6}) vertices for the Block Graph Deletion problem. This is a first step to investigate polynomial kernels for deletion problems into non-trivial classes of graphs of bounded rank-width, but unbounded tree-width. Our result also implies that Chordal Vertex Deletion admits a polynomial-size kernel on diamond-free graphs. For the kernelization and its analysis, we introduce the notion of 'complete degree' of a vertex. We believe that the underlying idea can be potentially applied to other problems. We also prove that the Block Graph Deletion problem can be solved in time 10^{k} * n^{O(1)}.

Cite As Get BibTex

Eun Jung Kim and O-joung Kwon. A Polynomial Kernel for Block Graph Deletion. In 10th International Symposium on Parameterized and Exact Computation (IPEC 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 43, pp. 270-281, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015) https://doi.org/10.4230/LIPIcs.IPEC.2015.270

Author Details

Eun Jung Kim
O-joung Kwon

References

  1. Hans L. Bodlaender. On disjoint cycles. In Proceedings of the 17th International Workshop, WG'91, pages 230-238, London, UK, UK, 1992. Springer-Verlag. Google Scholar
  2. Yixin Cao, Jianer Chen, and Yang Liu. On feedback vertex set: New measure and new structures. Algorithmica, pages 1-24, 2014. Google Scholar
  3. Yixin Cao and Dániel Marx. Chordal editing is fixed-parameter tractable. In 31st International Symposium on Theoretical Aspects of Computer Science, volume 25 of LIPIcs. Leibniz Int. Proc. Inform., pages 214-225. Schloss Dagstuhl. Leibniz-Zent. Inform., Wadern, 2014. Google Scholar
  4. Jianer Chen, Fedor V. Fomin, Yang Liu, Songjian Lu, and Yngve Villanger. Improved algorithms for feedback vertex set problems. J. Comput. System Sci., 74(7):1188-1198, 2008. Google Scholar
  5. Bruno Courcelle. The monadic second-order logic of graphs. I. Recognizable sets of finite graphs. Inform. and Comput., 85(1):12-75, 1990. Google Scholar
  6. Bruno Courcelle, Johann A. Makowsky, and Udi Rotics. Linear time solvable optimization problems on graphs of bounded clique-width. Theory Comput. Syst., 33(2):125-150, 2000. Google Scholar
  7. Marek Cygan, Jesper Nederlof, Marcin Pilipczuk, Michał Pilipczuk, Johan M. M. van Rooij, and Jakub Onufry Wojtaszczyk. Solving connectivity problems parameterized by treewidth in single exponential time (extended abstract). In 2011 IEEE 52nd Annual Symp. on Foundations of Computer Science (FOCS'11), pages 150-159. IEEE CS, 2011. Google Scholar
  8. Marek Cygan, Marcin Pilipczuk, Michał Pilipczuk, and Jakub Onufry Wojtaszczyk. Subset feedback vertex set is fixed-parameter tractable. SIAM Journal on Discrete Mathematics, 27(1):290-309, 2013. Google Scholar
  9. Marek Cygan, Marcin Pilipczuk, Michał Pilipczuk, and JakubOnufry Wojtaszczyk. An improved FPT algorithm and quadratic kernel for pathwidth one vertex deletion. In Venkatesh Raman and Saket Saurabh, editors, Parameterized and Exact Computation, volume 6478 of Lecture Notes in Computer Science, pages 95-106. Springer Berlin Heidelberg, 2010. Google Scholar
  10. Frank Dehne, Michael Fellows, Michael Langston, Frances Rosamond, and Kim Stevens. An O(2^O(k)n³) fpt algorithm for the undirected feedback vertex set problem. Theory of Computing Systems, 41(3):479-492, 2007. Google Scholar
  11. Rod Downey and Michael Fellows. Fixed-parameter tractability and completeness. III. Some structural aspects of the W hierarchy. In Complexity theory, pages 191-225. Cambridge Univ. Press, Cambridge, 1993. Google Scholar
  12. Fedor V. Fomin, Daniel Lokshtanov, Neeldhara Misra, and Saket Saurabh. Planar F-deletion: Approximation, kernelization and optimal FPT algorithms. In 53rd Annual IEEE Symposium on Foundations of Computer Science, FOCS 2012, New Brunswick, NJ, USA, October 20-23, 2012, pages 470-479, 2012. Google Scholar
  13. T. Gallai. Maximum-minimum Sätze und verallgemeinerte Faktoren von Graphen. Acta Math. Acad. Sci. Hungar., 12:131-173, 1961. Google Scholar
  14. Jiong Guo, Jens Gramm, Falk Hüffner, Rolf Niedermeier, and Sebastian Wernicke. Compression-based fixed-parameter algorithms for feedback vertex set and edge bipartization. Journal of Computer and System Sciences, 72(8):1386-1396, 2006. Google Scholar
  15. John Hopcroft and Robert Tarjan. Algorithm 447: Efficient algorithms for graph manipulation. Commun. ACM, 16(6):372-378, June 1973. Google Scholar
  16. Iyad Kanj, Michael Pelsmajer, and Marcus Schaefer. Parameterized algorithms for feedback vertex set. In Parameterized and Exact Computation, volume 3162 of LNCS, pages 235-247. Springer, 2004. Google Scholar
  17. Eun Jung Kim and O-joung Kwon. A polynomial kernel for Block Graph Vertex Deletion, 2015. arXiv.org:abs:1506.08477. Google Scholar
  18. Eun Jung Kim, Alexander Langer, Christophe Paul, Felix Reidl, Peter Rossmanith, Ignasi Sau, and Somnath Sikdar. Linear kernels and single-exponential algorithms via protrusion decompositions. In Automata, Languages, and Programming - 40th Int'l Colloquium, ICALP 2013, Proceedings, Part I, pages 613-624, 2013. Google Scholar
  19. Tomasz Kociumaka and Marcin Pilipczuk. Faster deterministic feedback vertex set. Information Processing Letters, 114(10):556-560, 2014. Google Scholar
  20. Dániel Marx. Chordal deletion is fixed-parameter tractable. Algorithmica, 57(4):747-768, 2010. Google Scholar
  21. Neeldhara Misra, Geevarghese Philip, Venkatesh Raman, and Saket Saurabh. On parameterized independent feedback vertex set. Theoretical Computer Science, 461(0):65-75, 2012. 17th International Computing and Combinatorics Conference (COCOON 2011). Google Scholar
  22. Sang-il Oum. Rank-width and vertex-minors. J. Combin. Theory Ser. B, 95(1):79-100, 2005. Google Scholar
  23. Venkatesh Raman, Saket Saurabh, and C. R. Subramanian. Faster fixed parameter tractable algorithms for undirected feedback vertex set. In Algorithms and computation, volume 2518 of Lecture Notes in Comput. Sci., pages 241-248. Springer, Berlin, 2002. Google Scholar
  24. Bruce Reed, Kaleigh Smith, and Adrian Vetta. Finding odd cycle transversals. Operations Research Letters, 32(4):299-301, 2004. Google Scholar
  25. Neil Robertson and P. D. Seymour. Graph minors. II. Algorithmic aspects of tree-width. J. Algorithms, 7(3):309-322, 1986. Google Scholar
  26. Stéphan Thomassé. A quadratic kernel for feedback vertex set. In Proceedings of the Twentieth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 115-119. ACM/SIAM, 2009. Google Scholar
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