Linear-time algorithms for scattering number and hamilton-connectivity of interval graphs

Haitze J. Broersma, Jiri Fiala, Petr A. Golovach, Tomas Kaiser, Daniël Paulusma, Andrzej Proskurowski

  • 3 Citations

Abstract

We prove that for all integers k<0 an interval graph is -(k+1)-Hamilton-connected if and only if its scattering number is at most k. This complements a previously known fact that an interval graph has a nonnegative scattering number if and only if it contains a Hamilton cycle, as well as a characterization of interval graphs with positive scattering numbers in terms of the minimum size of a path cover. 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 previously best-known O(n^3) time bound for solving this problem. 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.
Original languageUndefined
Pages (from-to)282-299
Number of pages18
JournalJournal of graph theory
Volume79
Issue number4
DOIs
StatePublished - Aug 2015

Fingerprint

Interval graphs
Scattering
If and only if
Hamilton cycle
Complement
Non-negative
Cover
Path
Integer
Computing

Keywords

  • MSC-05C
  • METIS-314967
  • IR-98274
  • EWI-26316

Cite this

Broersma, H. J., Fiala, J., Golovach, P. A., Kaiser, T., Paulusma, D., & Proskurowski, A. (2015). Linear-time algorithms for scattering number and hamilton-connectivity of interval graphs. Journal of graph theory, 79(4), 282-299. DOI: 10.1002/jgt.21832

Broersma, Haitze J.; Fiala, Jiri; Golovach, Petr A.; Kaiser, Tomas; Paulusma, Daniël; Proskurowski, Andrzej / Linear-time algorithms for scattering number and hamilton-connectivity of interval graphs.

In: Journal of graph theory, Vol. 79, No. 4, 08.2015, p. 282-299.

Research output: Scientific - peer-reviewArticle

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Broersma, HJ, Fiala, J, Golovach, PA, Kaiser, T, Paulusma, D & Proskurowski, A 2015, 'Linear-time algorithms for scattering number and hamilton-connectivity of interval graphs' Journal of graph theory, vol 79, no. 4, pp. 282-299. DOI: 10.1002/jgt.21832

Linear-time algorithms for scattering number and hamilton-connectivity of interval graphs. / Broersma, Haitze J.; Fiala, Jiri; Golovach, Petr A.; Kaiser, Tomas; Paulusma, Daniël; Proskurowski, Andrzej.

In: Journal of graph theory, Vol. 79, No. 4, 08.2015, p. 282-299.

Research output: Scientific - peer-reviewArticle

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AU - Broersma,Haitze J.

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AU - Paulusma,Daniël

AU - Proskurowski,Andrzej

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N2 - We prove that for all integers k<0 an interval graph is -(k+1)-Hamilton-connected if and only if its scattering number is at most k. This complements a previously known fact that an interval graph has a nonnegative scattering number if and only if it contains a Hamilton cycle, as well as a characterization of interval graphs with positive scattering numbers in terms of the minimum size of a path cover. 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 previously best-known O(n^3) time bound for solving this problem. 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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Broersma HJ, Fiala J, Golovach PA, Kaiser T, Paulusma D, Proskurowski A. Linear-time algorithms for scattering number and hamilton-connectivity of interval graphs. Journal of graph theory. 2015 Aug;79(4):282-299. Available from, DOI: 10.1002/jgt.21832