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Parameterized Edge Hamiltonicity

  • Michael Lampis
  • Kazuhisa Makino
  • Valia MitsouEmail author
  • Yushi Uno
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8747)

Abstract

We study the parameterized complexity of the classical Edge Hamiltonian Path problem and give several fixed-parameter tractability results. First, we settle an open question of Demaine et al. by showing that Edge Hamiltonian Path is FPT parameterized by vertex cover, and that it also admits a cubic kernel. We then show fixed-parameter tractability even for a generalization of the problem to arbitrary hypergraphs, parameterized by the size of a (supplied) hitting set. We also consider the problem parameterized by treewidth or clique-width. Surprisingly, we show that the problem is FPT for both of these standard parameters, in contrast to its vertex version, which is W[1]-hard for clique-width. Our technique, which may be of independent interest, relies on a structural characterization of clique-width in terms of treewidth and complete bipartite subgraphs due to Gurski and Wanke.

References

  1. 1.
    Bertossi, A.A.: The edge Hamiltonian path problem is NP-complete. Inf. Process. Lett. 13(4), 157–159 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bodlaender, H.L., Drange, P.G., Dregi, M.S., Fomin, F.V., Lokshtanov, D., Pilipczuk, M.: An \(O(c^k n)\) 5-Approximation algorithm for treewidth. In: FOCS, pp. 499–508. IEEE Computer Society (2013)Google Scholar
  3. 3.
    Bodlaender, H.L., Koster, A.M.: Combinatorial optimization on graphs of bounded treewidth. Comput. J. 51(3), 255–269 (2008)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Brualdi, R.A., Shanny, R.F.: Hamiltonian line graphs. J. Graph Theory 5(3), 307–314 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Catlin, P.A.: Supereulerian graphs: a survey. J. Graph Theory 16(2), 177–196 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Chartrand, G.: On Hamiltonian line-graphs. Trans. Am. Math. Soc. 134, 559–566 (1968)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Chen, Z.-H., Lai, H.-J., Li, X., Li, D., Mao, J.: Eulerian subgraphs in 3-edge-connected graphs and Hamiltonian line graphs. J. Graph Theory 42(4), 308–319 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Clark, L.: On Hamiltonian line graphs. J. Graph Theory 8(2), 303–307 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Cormen, T.H., Leiserson, C.E., Rivest, R.L., Stein, C., et al.: Introduction to Algorithms, vol. 2. MIT press Cambridge, Cambridge (2001)zbMATHGoogle Scholar
  10. 10.
    Corneil, D.G., Rotics, U.: On the relationship between clique-width and treewidth. SIAM J. Comput. 34(4), 825–847 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Courcelle, B.: The monadic second-order logic of graphs. I. recognizable sets of finite graphs. Inf. Comput. 85(1), 12–75 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Courcelle, B., Engelfriet, J.: Graph Structure and Monadic Second-Order Logic: A Language-Theoretic Approach. Cambridge University Press, New York (2012)CrossRefGoogle Scholar
  13. 13.
    Courcelle, B., Makowsky, J.A., Rotics, U.: Linear time solvable optimization problems on graphs of bounded clique-width. Theory Comput. Syst. 33(2), 125–150 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Demaine, E.D., Demaine, M.L., Harvey, N.J.A., Uehara, R., Uno, T., Uno, Y.: UNO is hard, even for a single player. Theor. Comput. Sci. 521, 51–61 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Dey, P., Goyal, P., Misra, N.: UNO gets easier for a single player. In: Ferro, A., Luccio, F., Widmayer, P. (eds.) FUN 2014. LNCS, vol. 8496, pp. 147–157. Springer, Heidelberg (2014)CrossRefGoogle Scholar
  16. 16.
    Downey, R.G., Fellows, M.R.: Parameterized Complexity. Springer, New York (1999)CrossRefGoogle Scholar
  17. 17.
    Espelage, W., Gurski, F., Wanke, E.: How to solve NP-hard graph problems on clique-width bounded graphs in polynomial time. In: Brandstädt, A., Le, V.B. (eds.) WG 2001. LNCS, vol. 2204, pp. 117–128. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  18. 18.
    Flum, J., Grohe, M.: Parameterized Complexity Theory, vol. 3. Springer, Berlin (2006)Google Scholar
  19. 19.
    Fomin, F.V., Golovach, P.A., Lokshtanov, D., Saurabh, S.: Clique-width: on the price of generality. In: Mathieu, C. (ed.) SODA, pp. 825–834. SIAM, Philadelphia (2009)Google Scholar
  20. 20.
    Gurski, F., Wanke, E.: The tree-width of clique-width bounded graphs without \(K_{n,n}\). In: Brandes, U., Wagner, D. (eds.) WG 2000. LNCS, vol. 1928, pp. 196–205. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  21. 21.
    Gurski, F., Wanke, E.: Line graphs of bounded clique-width. Discrete Math. 307(22), 2734–2754 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Harary, F., Nash-Williams, C.S.J.: On Eulerian and Hamiltonian graphs and line graphs. Can. Math. Bull. 8, 701–709 (1965)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Hliněnỳ, P., Oum, S.-I., Seese, D., Gottlob, G.: Width parameters beyond tree-width and their applications. Comput. J. 51(3), 326–362 (2008)Google Scholar
  24. 24.
    Lai, H.-J.: Eulerian subgraphs containing given vertices and hamiltonian line graphs. Discrete Math. 178(1), 93–107 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Lai, T.-H., Wei, S.-S.: The edge Hamiltonian path problem is NP-complete for bipartite graphs. Inf. Process. Lett. 46(1), 21–26 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Niedermeier, R.: Invitation to Fixed-Parameter Algorithms: Oxford Lecture Series in Mathematics and Its Applications. OUP, Oxford (2006)CrossRefGoogle Scholar
  27. 27.
    Oum, S.-I., Seymour, P.: Approximating clique-width and branch-width. J. Comb. Theory Ser. B 96(4), 514–528 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Razgon, I., Petke, J.: Cliquewidth and knowledge compilation. In: Järvisalo, M., Van Gelder, A. (eds.) SAT 2013. LNCS, vol. 7962, pp. 335–350. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  29. 29.
    Ryjácek, Z., Woeginger, G.J., Xiong, L.: Hamiltonian index is NP-complete. Discrete Appl. Math. 159(4), 246–250 (2011)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Michael Lampis
    • 1
  • Kazuhisa Makino
    • 1
  • Valia Mitsou
    • 2
    Email author
  • Yushi Uno
    • 3
  1. 1.Research Institute for Mathematical SciencesKyoto UniversityKyotoJapan
  2. 2.JST Erato Minato Discrete Structure Manipulation System ProjectSapporoJapan
  3. 3.Department of Mathematics and Information Sciences, Graduate School of ScienceOsaka Prefecture UniversityOsakaJapan

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