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Efficient Enumeration of Maximal k-Degenerate Subgraphs in a Chordal Graph

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Computing and Combinatorics (COCOON 2017)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10392))

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Abstract

In this paper, we consider the problem of listing the maximal k-degenerate induced subgraphs of a chordal graph, and propose an output-sensitive algorithm using delay \(O(m\cdot \omega (G))\) for any n-vertex chordal graph with m edges, where \(\omega (G) \le n\) is the maximum size of a clique in G. The problem generalizes that of enumerating maximal independent sets and maximal induced forests, which correspond to respectively 0-degenerate and 1-degenerate subgraphs.

M.M. Kanté is supported by French Agency for Research under the GraphEN project (ANR-15-CE-0009)

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References

  1. Alon, N., Kahn, J., Seymour, P.D.: Large induced degenerate subgraphs. Graph. Combin. 3(1), 203–211 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bauer, R., Krug, M., Wagner, D.: Enumerating and generating labeled \(k\)-degenerate graphs. In: 2010 Proceedings of the Seventh Workshop on Analytic Algorithmics and Combinatorics, pp. 90–98. SIAM, Philadelphia (2010)

    Google Scholar 

  3. Blair, J.R., Peyton, B.: An introduction to chordal graphs and clique trees. In: George, A., Gilbert, J.R., Liu, J.W.H. (eds.) Graph Theory and Sparse Matrix Computation, pp. 1–29. Springer, New York (1993)

    Google Scholar 

  4. Chandran, L.S.: A linear time algorithm for enumerating all the minimum and minimal separators of a chordal graph. In: Wang, J. (ed.) COCOON 2001. LNCS, vol. 2108, pp. 308–317. Springer, Heidelberg (2001). doi:10.1007/3-540-44679-6_34

    Chapter  Google Scholar 

  5. Conte, A., Grossi, R., Marino, A., Versari, L.: Sublinear-space bounded-delay enumeration for massive network analytics: Maximal cliques. In: ICALP (2016)

    Google Scholar 

  6. Diestel, R.: Graph Theory (Graduate Texts in Mathematics). Springer, New York (2005)

    Google Scholar 

  7. Eiter, T., Makino, K., Gottlob, G.: Computational aspects of monotone dualization: a brief survey. Discrete Appl. Math. 156(11), 2035–2049 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  8. Enright, J., Kondrak, G.: The application of chordal graphs to inferring phylogenetic trees of languages. In: Fifth International Joint Conference on Natural Language Processing, IJCNLP, pp. 545–552 (2011)

    Google Scholar 

  9. Eppstein, D., Löffler, M., Strash, D.: Listing all maximal cliques in sparse graphs in near-optimal time. In: Cheong, O., Chwa, K.-Y., Park, K. (eds.) ISAAC 2010. LNCS, vol. 6506, pp. 403–414. Springer, Heidelberg (2010). doi:10.1007/978-3-642-17517-6_36

    Chapter  Google Scholar 

  10. Galinier, P., Habib, M., Paul, C.: Chordal graphs and their clique graphs. In: Nagl, M. (ed.) WG 1995. LNCS, vol. 1017, pp. 358–371. Springer, Heidelberg (1995). doi:10.1007/3-540-60618-1_88

    Chapter  Google Scholar 

  11. Gavril, F.: The intersection graphs of subtrees in trees are exactly the chordal graphs. J. Comb. Theor. Ser. B 16(1), 47–56 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  12. Kuramochi, M., Karypis, G.: Frequent subgraph discovery. In: Proceedings IEEE International Conference on Data Mining, pp. 313–320. IEEE (2001)

    Google Scholar 

  13. Lee, V.E., Ruan, N., Jin, R., Aggarwal, C.: A survey of algorithms for dense subgraph discovery. In: Aggarwal, C.C., Wang, H. (eds.) Managing and Mining Graph Data, pp. 303–336. Springer, Heidelberg (2010)

    Chapter  Google Scholar 

  14. Lukot’ka, R., Mazák, J., Zhu, X.: Maximum 4-degenerate subgraph of a planar graph. Electron. J. Comb. 22(1), P1–11 (2015)

    MathSciNet  MATH  Google Scholar 

  15. Nešetřil, J., Ossona de Mendez, P.: Sparsity: Graphs, Structures, and Algorithms. Algorithms and Combinatorics, vol. 28. Springer, Heidelberg (2012)

    Google Scholar 

  16. Pilipczuk, M., Pilipczuk, M.: Finding a maximum induced degenerate subgraph faster than 2n. In: Thilikos, D.M., Woeginger, G.J. (eds.) IPEC 2012. LNCS, vol. 7535, pp. 3–12. Springer, Heidelberg (2012). doi:10.1007/978-3-642-33293-7_3

    Chapter  Google Scholar 

  17. Rose, D.J., Tarjan, R.E., Lueker, G.S.: Algorithmic aspects of vertex elimination on graphs. SIAM J. Comput. 5(2), 266–283 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  18. Tarjan, R.E., Yannakakis, M.: Simple linear-time algorithms to test chordality of graphs, test acyclicity of hypergraphs, and selectively reduce acyclic hypergraphs. SIAM J. Comput. 13(3), 566–579 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  19. Ugander, J., Karrer, B., Backstrom, L., Marlow, C.: The anatomy of the facebook social graph. arXiv preprint arXiv: 1111.4503 (2011)

  20. Wasa, K.: Enumeration of enumeration algorithms. arXiv preprint arXiv:1605.05102 (2016)

  21. Wasa, K., Arimura, H., Uno, T.: Efficient enumeration of induced subtrees in a K-degenerate graph. In: Ahn, H.-K., Shin, C.-S. (eds.) ISAAC 2014. LNCS, vol. 8889, pp. 94–102. Springer, Cham (2014). doi:10.1007/978-3-319-13075-0_8

    Google Scholar 

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Authors and Affiliations

  1. Università di Pisa, Pisa, Italy

    Alessio Conte

  2. Université Clermont Auvergne, LIMOS, CNRS, Aubière, France

    Mamadou Moustapha Kanté

  3. Kumamoto University, Kumamoto, Japan

    Yota Otachi

  4. National Institute of Informatics, Tokyo, Japan

    Takeaki Uno & Kunihiro Wasa

Authors
  1. Alessio Conte
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  2. Mamadou Moustapha Kanté
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  3. Yota Otachi
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  4. Takeaki Uno
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  5. Kunihiro Wasa
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Corresponding author

Correspondence to Alessio Conte .

Editor information

Editors and Affiliations

  1. Department of Computing, Hong Kong Polytechnic University, Hong Kong, China

    Yixin Cao

  2. Texas A&M University, College Station, Texas, USA

    Jianer Chen

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Conte, A., Kanté, M.M., Otachi, Y., Uno, T., Wasa, K. (2017). Efficient Enumeration of Maximal k-Degenerate Subgraphs in a Chordal Graph. In: Cao, Y., Chen, J. (eds) Computing and Combinatorics. COCOON 2017. Lecture Notes in Computer Science(), vol 10392. Springer, Cham. https://doi.org/10.1007/978-3-319-62389-4_13

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  • DOI: https://doi.org/10.1007/978-3-319-62389-4_13

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  • Print ISBN: 978-3-319-62388-7

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