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 journal › Article › Academic › peer-review

7 Citations (Scopus)
12 Downloads (Pure)

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.
@article{3403f2b246d84b3b9c9667385ebe9e19,
title = "Linear-time algorithms for scattering number and hamilton-connectivity of interval graphs",
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.",
keywords = "MSC-05C, METIS-314967, IR-98274, EWI-26316",
author = "Broersma, {Haitze J.} and Jiri Fiala and Golovach, {Petr A.} and Tomas Kaiser and Dani{\"e}l Paulusma and Andrzej Proskurowski",
note = "eemcs-eprint-26316",
year = "2015",
month = "8",
doi = "10.1002/jgt.21832",
language = "Undefined",
volume = "79",
pages = "282--299",
journal = "Journal of graph theory",
issn = "0364-9024",
publisher = "Wiley-Liss Inc.",
number = "4",

}

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 journal › Article › Academic › peer-review

TY - JOUR

T1 - Linear-time algorithms for scattering number and hamilton-connectivity of interval graphs

AU - Broersma, Haitze J.

AU - Fiala, Jiri

AU - Golovach, Petr A.

AU - Kaiser, Tomas

AU - Paulusma, Daniël

AU - Proskurowski, Andrzej

N1 - eemcs-eprint-26316

PY - 2015/8

Y1 - 2015/8

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.

KW - MSC-05C

KW - METIS-314967

KW - IR-98274

KW - EWI-26316

U2 - 10.1002/jgt.21832

DO - 10.1002/jgt.21832

M3 - Article

VL - 79

SP - 282

EP - 299

JO - Journal of graph theory

JF - Journal of graph theory

SN - 0364-9024

IS - 4

ER -

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. https://doi.org/10.1002/jgt.21832

Access to Document