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Fully Leafed Induced Subtrees

  • Alexandre Blondin Massé
  • Julien de Carufel
  • Alain Goupil
  • Mélodie Lapointe
  • Émile Nadeau
  • Élise VandommeEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10979)

Abstract

We consider the problem \(\mathrm {LIS}\) of deciding whether there exists an induced subtree with exactly \(i \le n\) vertices and \(\ell \) leaves in a given graph G with n vertices. We study the associated optimization problem, that consists in computing the maximal number of leaves, denoted by \(L_G(i)\), realized by an induced subtree with i vertices, for \(0 \le i \le n\). We begin by proving that the \(\mathrm {LIS}\) problem is NP-complete in general. Then, we describe a nontrivial branch and bound algorithm that computes the function \(L_G\) for any simple graph G. In the special case where G is a tree of maximum degree \(\varDelta \), we provide a \(\mathcal {O}(n^3\varDelta )\) time and \(\mathcal {O}(n^2)\) space algorithm to compute the function \(L_G\).

References

  1. 1.
    Blondin Massé, A., de Carufel, J., Goupil, A., Lapointe, M., Nadeau, É., Vandomme, É.: Fully leafed induced subtrees (2017). arXiv.org/abs/1709.09808
  2. 2.
    Blondin Massé, A., de Carufel, J., Goupil, A., Samson, M.: Fully leafed tree-like polyominoes and polycubes. In: Brankovic, L., Ryan, J., Smyth, W.F. (eds.) IWOCA 2017. LNCS, vol. 10765, pp. 206–218. Springer, Cham (2018).  https://doi.org/10.1007/978-3-319-78825-8_17CrossRefGoogle Scholar
  3. 3.
    Blondin Massé, A., Nadeau, É.: Fully leafed induced subtrees, GitHub Repository. https://github.com/enadeau/fully-leafed-induced-subtrees
  4. 4.
    Bodlaender, H.L.: On linear time minor tests and depth first search. In: Dehne, F., Sack, J.-R., Santoro, N. (eds.) WADS 1989. LNCS, vol. 382, pp. 577–590. Springer, Heidelberg (1989).  https://doi.org/10.1007/3-540-51542-9_48CrossRefzbMATHGoogle Scholar
  5. 5.
    Boukerche, A., Cheng, X., Linus, J.: A performance evaluation of a novel energy-aware data-centric routing algorithm in wireless sensor networks. Wirel. Netw. 11(5), 619–635 (2005)CrossRefGoogle Scholar
  6. 6.
    Chen, S., Ljubić, I., Raghavan, S.: The generalized regenerator location problem. INFORMS J. Comput. 27(2), 204–220 (2015)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Deepak, A., Fernández-Baca, D., Tirthapura, S., Sanderson, M.J., McMahon, M.M.: EvoMiner: frequent subtree mining in phylogenetic databases. Knowl. Inf. Syst. 41(3), 559–590 (2014)CrossRefGoogle Scholar
  8. 8.
    Diestel, R.: Graph Theory. Graduate Texts in Mathematics, vol. 173, 4th edn. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  9. 9.
    Downey, R.G., Fellows, M.R.: Fixed-parameter tractability and completeness II: on completeness for W[1]. Theoret. Comput. Sci. 141(1), 109–131 (1995)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Downey, R.G., Fellows, M.R.: Parameterized computational feasibility. In: Clote, P., Remmel, J.B. (eds.) Feasible Mathematics II, pp. 219–244. Birkhäuser Boston, Boston (1995)CrossRefGoogle Scholar
  11. 11.
    Downey, R.G., Fellows, M.R.: Parameterized Complexity. Monographs in Computer Science. Springer, New York (1999).  https://doi.org/10.1007/978-1-4612-0515-9CrossRefzbMATHGoogle Scholar
  12. 12.
    Erdős, P., Saks, M., Sós, V.T.: Maximum induced trees in graphs. J. Combin. Theory Ser. B 41(1), 61–79 (1986)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Garey, M.R., Johnson, D.S.: Computers and Intractability. W. H. Freeman and Co., San Francisco (1979)zbMATHGoogle Scholar
  14. 14.
    Payan, C., Tchuente, M., Xuong, N.H.: Arbres avec un nombre maximum de sommets pendants (Trees with a maximal number of vertices with degree 1). Discrete Math. 49(3), 267–273 (1984)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Székely, L.A., Wang, H.: On subtrees of trees. Adv. Appl. Math. 34(1), 138–155 (2005)MathSciNetCrossRefGoogle Scholar
  16. 16.
    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).  https://doi.org/10.1007/978-3-319-13075-0_8CrossRefGoogle Scholar
  17. 17.
    Zaki, M.J.: Efficiently mining frequent trees in a forest. In: Proceedings of the Eighth ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, KDD 2002, pp. 71–80. ACM, New York (2002)Google Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Alexandre Blondin Massé
    • 1
  • Julien de Carufel
    • 2
  • Alain Goupil
    • 2
  • Mélodie Lapointe
    • 1
  • Émile Nadeau
    • 1
  • Élise Vandomme
    • 1
    Email author
  1. 1.Laboratoire de Combinatoire et d’Informatique MathématiqueUniversité du Québec à MontréalMontrealCanada
  2. 2.Laboratoire Interdisciplinaire de Recherche en Imagerie et en CombinatoireUniversité du Québec à Trois-RivièresTrois-RivièresCanada

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