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

Research output: Contribution to journalArticleAcademicpeer-review

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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
Publication statusPublished - Aug 2015

Keywords

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

Cite this

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. 2015 ; Vol. 79, No. 4. pp. 282-299.
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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.",
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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. https://doi.org/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: Contribution to journalArticleAcademicpeer-review

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AU - Broersma, Haitze J.

AU - Fiala, Jiri

AU - Golovach, Petr A.

AU - Kaiser, Tomas

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.

AB - 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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