Skip to main content
SpringerLink
Log in

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

  • Conference paper
Graph-Theoretic Concepts in Computer Science (WG 2013)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 8165))

Included in the following conference series:

  • 905 Accesses

Abstract

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.

Paper supported by Royal Society Joint Project Grant JP090172.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Arikati, S.R., Pandu Rangan, C.: Linear algorithm for optimal path cover problem on interval graphs. Inf. Proc. Let. 35, 149–153 (1990)

    Article  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  3. Asdre, K., Nikolopoulos, S.D.: The 1-fixed-endpoint path cover problem is polynomial on interval graphs. Algorithmica 58, 679–710 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bertossi, A.A.: Finding hamiltonian circuits in proper interval graphs. Inf. Proc. Let. 17, 97–101 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bertossi, A.A., Bonucelli, M.A.: Hamilton circuits in interval graph generalizations. Inf. Proc. Let. 23, 195–200 (1986)

    Article  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Chapter  Google Scholar 

  9. Chen, C., Chang, C.-C., Chang, G.J.: Proper interval graphs and the guard problem. Disc. Math. 170, 223–230 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  10. Chvátal, V.: Tough graphs and hamiltonian circuits. Disc. Math. 5, 215–228 (1973)

    Article  MATH  Google Scholar 

  11. Damaschke, P.: Paths in interval graphs and circular arc graphs. Disc. Math. 112, 49–64 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  12. Dean, A.M.: The computational complexity of deciding hamiltonian-connectedness. Congr. Num. 93, 209–214 (1993)

    MathSciNet  MATH  Google Scholar 

  13. Deogun, J.S., Kratsch, D., Steiner, G.: 1-tough cocomparability graphs are Hamiltonian. Disc. Math. 170, 99–106 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  14. Deogun, J.S., Steiner, G.: Polynomial algorithms for hamiltonian cycle in cocomparability graphs. SIAM J. Comp. 23, 520–552 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  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. 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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  18. Jung, H.A.: On a class of posets and the corresponding comparability graphs. J. Comb. Th. B 24, 125–133 (1978)

    Article  MATH  Google Scholar 

  19. Keil, J.M.: Finding hamiltonian circuits in interval graphs. Inf. Proc. Let. 20, 201–206 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  20. Kratsch, D., Kloks, T., Müller, H.: Measuring the vulnerability for classes of intersection graphs. Disc. Appl. Math. 77, 259–270 (1997)

    Article  MATH  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  22. Li, P., Wu, Y.: A linear time algorithm for solving the 1-fixed-endpoint path cover problem on interval graphs, draft

    Google Scholar 

  23. Lehel, J.: The path partition of cocomparability graphs (1991) (manuscript)

    Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Faculty of EEMCS, University of Twente, Enschede, The Netherlands

    Hajo Broersma

  2. Department of Applied Mathematics, Charles University, Prague, Czech Republic

    Jiří Fiala

  3. Institute of Computer Science, University of Bergen, Norway

    Petr A. Golovach

  4. Department of Mathematics, University of West Bohemia, Plzeň, Czech Republic

    Tomáš Kaiser

  5. School of Engineering and Computing Sciences, Durham University, UK

    Daniël Paulusma

  6. Department of Computer Science, University of Oregon, Eugene, USA

    Andrzej Proskurowski

Authors
  1. Hajo Broersma
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jiří Fiala
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Petr A. Golovach
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Tomáš Kaiser
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Daniël Paulusma
    View author publications

    You can also search for this author in PubMed Google Scholar

  6. Andrzej Proskurowski
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Institute of Computer Science, University of Rostock, Albert-Einstein-Straße 22, 18059, Rostock, Germany

    Andreas Brandstädt

  2. Department of Computer Science, University of Kiel, Olshausenstraße 40, 24098, Kiel, Germany

    Klaus Jansen

  3. Institute of Theoretical Computer Science, University of Lübeck, Ratzeburger Allee 160, 23538, Lübeck, Germany

    Rüdiger Reischuk

Rights and permissions

Reprints and permissions

Copyright information

© 2013 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Broersma, H., Fiala, J., Golovach, P.A., Kaiser, T., Paulusma, D., Proskurowski, A. (2013). Linear-Time Algorithms for Scattering Number and Hamilton-Connectivity of Interval Graphs. In: Brandstädt, A., Jansen, K., Reischuk, R. (eds) Graph-Theoretic Concepts in Computer Science. WG 2013. Lecture Notes in Computer Science, vol 8165. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-45043-3_12

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-45043-3_12

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-45042-6

  • Online ISBN: 978-3-642-45043-3

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation