Linear-Time Algorithms for Scattering Number and Hamilton-Connectivity of Interval Graphs

  • Hajo Broersma
  • Jiří Fiala
  • Petr A. Golovach
  • Tomáš Kaiser
  • Daniël Paulusma
  • Andrzej Proskurowski
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8165)


We show that for all k ≤ − 1 an interval graph is − (k + 1)-Hamilton-connected if and only if its scattering number is at most k. We also give an O(n + m) time algorithm for computing the scattering number of an interval graph with n vertices and m edges, which improves the O(n 3) time bound of Kratsch, Kloks and Müller. As a consequence of our two results the maximum k for which an interval graph is k-Hamilton-connected can be computed in O(n + m) time.


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  1. 1.
    Arikati, S.R., Pandu Rangan, C.: Linear algorithm for optimal path cover problem on interval graphs. Inf. Proc. Let. 35, 149–153 (1990)CrossRefMATHGoogle Scholar
  2. 2.
    Asdre, K., Nikolopoulos, S.D.: A polynomial solution to the k-fixed-endpoint path cover problem on proper interval graphs. Theor. Comp. Sci. 411, 967–975 (2010)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Asdre, K., Nikolopoulos, S.D.: The 1-fixed-endpoint path cover problem is polynomial on interval graphs. Algorithmica 58, 679–710 (2010)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Bertossi, A.A.: Finding hamiltonian circuits in proper interval graphs. Inf. Proc. Let. 17, 97–101 (1983)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Bertossi, A.A., Bonucelli, M.A.: Hamilton circuits in interval graph generalizations. Inf. Proc. Let. 23, 195–200 (1986)CrossRefMATHGoogle Scholar
  6. 6.
    Booth, K.S., Lueker, G.S.: Testing for the consecutive ones property, interval graphs, and graph planarity using pq-tree algorithms. J. Com. Sys. Sci. 13, 335–379 (1976)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Chang, M.-S., Peng, S.-L., Liaw, J.-L.: Deferred-query: An efficient approach for some problems on interval graphs. Networks 34, 1–10 (1999)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Chen, C., Chang, C.-C.: Connected proper interval graphs and the guard problem in spiral polygons. In: Deza, M., Manoussakis, I., Euler, R. (eds.) CCS 1995. LNCS, vol. 1120, pp. 39–47. Springer, Heidelberg (1996)CrossRefGoogle Scholar
  9. 9.
    Chen, C., Chang, C.-C., Chang, G.J.: Proper interval graphs and the guard problem. Disc. Math. 170, 223–230 (1997)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Chvátal, V.: Tough graphs and hamiltonian circuits. Disc. Math. 5, 215–228 (1973)CrossRefMATHGoogle Scholar
  11. 11.
    Damaschke, P.: Paths in interval graphs and circular arc graphs. Disc. Math. 112, 49–64 (1993)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Dean, A.M.: The computational complexity of deciding hamiltonian-connectedness. Congr. Num. 93, 209–214 (1993)MathSciNetMATHGoogle Scholar
  13. 13.
    Deogun, J.S., Kratsch, D., Steiner, G.: 1-tough cocomparability graphs are Hamiltonian. Disc. Math. 170, 99–106 (1997)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Deogun, J.S., Steiner, G.: Polynomial algorithms for hamiltonian cycle in cocomparability graphs. SIAM J. Comp. 23, 520–552 (1994)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman & Co. Ltd. (1979)Google Scholar
  16. 16.
    Hung, R.-W., Chang, M.-S.: Linear-time certifying algorithms for the path cover and hamiltonian cycle problems on interval graphs. Appl. Math. Lett. 24, 648–652 (2011)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Ioannidou, K., Mertzios, G.B., Nikolopoulos, S.D.: The longest path problem has a polynomial solution on interval graphs. Algorithmica 61, 320–341 (2011)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Jung, H.A.: On a class of posets and the corresponding comparability graphs. J. Comb. Th. B 24, 125–133 (1978)CrossRefMATHGoogle Scholar
  19. 19.
    Keil, J.M.: Finding hamiltonian circuits in interval graphs. Inf. Proc. Let. 20, 201–206 (1985)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Kratsch, D., Kloks, T., Müller, H.: Measuring the vulnerability for classes of intersection graphs. Disc. Appl. Math. 77, 259–270 (1997)CrossRefMATHGoogle Scholar
  21. 21.
    Kužel, R., Ryjáček, Z., Vrána, P.: Thomassen’s conjecture implies polynomiality of 1-hamilton-connectedness in line graphs. J Graph Th. 69, 241–250 (2012)CrossRefMATHGoogle Scholar
  22. 22.
    Li, P., Wu, Y.: A linear time algorithm for solving the 1-fixed-endpoint path cover problem on interval graphs, draftGoogle Scholar
  23. 23.
    Lehel, J.: The path partition of cocomparability graphs (1991) (manuscript)Google Scholar
  24. 24.
    Manacher, G.K., Mankus, T.A., Smith, C.J.: An optimum Θ(nlogn) algorithm for finding a canonical hamiltonian path and a canonical hamiltonian circuit in a set of intervals. Inf. Proc. Let. 35, 205–211 (1990)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Shih, W.K., Chern, T.C., Hsu, W.L.: An O(n 2logn) time algorithm for the hamiltonian cycle problem on circular-arc graphs. SIAM J. Comp. 21, 1026–1046 (1992)MathSciNetCrossRefMATHGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Hajo Broersma
    • 1
  • Jiří Fiala
    • 2
  • Petr A. Golovach
    • 3
  • Tomáš Kaiser
    • 4
  • Daniël Paulusma
    • 5
  • Andrzej Proskurowski
    • 6
  1. 1.Faculty of EEMCSUniversity of TwenteEnschedeThe Netherlands
  2. 2.Department of Applied MathematicsCharles UniversityPragueCzech Republic
  3. 3.Institute of Computer ScienceUniversity of BergenNorway
  4. 4.Department of MathematicsUniversity of West BohemiaPlzeňCzech Republic
  5. 5.School of Engineering and Computing SciencesDurham UniversityUK
  6. 6.Department of Computer ScienceUniversity of OregonEugeneUSA

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