Independent sets in asteroidal triple-free graphs

Haitze J. Broersma, Ton Kloks, A.J.J. Kloks, Dieter Kratsch, Haiko Müller

Research output: Chapter in Book/Report/Conference proceedingConference contributionAcademic

9 Citations (Scopus)
104 Downloads (Pure)

Abstract

An asteroidal triple is a set of three vertices such that there is a path between any pair of them avoiding the closed neighborhood of the third. A graph is called AT-free if it does not have an asteroidal triple. We show that there is an O(n 2 · (¯m+1)) time algorithm to compute the maximum cardinality of an independent set for AT-free graphs, where n is the number of vertices and ¯m is the number of non edges of the input graph. Furthermore we obtain O(n 2 · (¯m+1)) time algorithms to solve the INDEPENDENT DOMINATING SET and the INDEPENDENT PERFECT DOMINATING SET problem on AT-free graphs. We also show how to adapt these algorithms such that they solve the corresponding problem for graphs with bounded asteroidal number in polynomial time. Finally we observe that the problems CLIQUE and PARTITION INTO CLIQUES remain NP-complete when restricted to AT-free graphs.
Original languageEnglish
Title of host publicationAutomata, Languages and Programming
Subtitle of host publication24th International Colloquium, ICALP '97 Bologna, Italy, July 7–11, 1997 Proceedings
EditorsPierpaolo Degano, Roberto Gorrieri, Alberto Marchetti-Spaccamela
PublisherSpringer
Pages760-770
Number of pages11
ISBN (Electronic)978-3-540-69194-5
ISBN (Print)978-3-540-63165-1
DOIs
Publication statusPublished - 1997
Event24th International Colloquium on Automata, Languages and Programming, ICALP 1997 - Bologna, Italy
Duration: 7 Jul 199711 Jul 1997
Conference number: 24

Publication series

NameLecture notes in computer science
PublisherSpringer
Volume1256
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349

Conference

Conference24th International Colloquium on Automata, Languages and Programming, ICALP 1997
Abbreviated titleICALP
CountryItaly
CityBologna
Period7/07/9711/07/97

Fingerprint

Independent Set
Graph in graph theory
Dominating Set
Cardinality
Polynomial time
NP-complete problem
Closed
Path

Keywords

  • IR-92411
  • METIS-140801

Cite this

Broersma, H. J., Kloks, T., Kloks, A. J. J., Kratsch, D., & Müller, H. (1997). Independent sets in asteroidal triple-free graphs. In P. Degano, R. Gorrieri, & A. Marchetti-Spaccamela (Eds.), Automata, Languages and Programming: 24th International Colloquium, ICALP '97 Bologna, Italy, July 7–11, 1997 Proceedings (pp. 760-770). (Lecture notes in computer science; Vol. 1256). Springer. https://doi.org/10.1007/3-540-63165-8_229
Broersma, Haitze J. ; Kloks, Ton ; Kloks, A.J.J. ; Kratsch, Dieter ; Müller, Haiko. / Independent sets in asteroidal triple-free graphs. Automata, Languages and Programming: 24th International Colloquium, ICALP '97 Bologna, Italy, July 7–11, 1997 Proceedings. editor / Pierpaolo Degano ; Roberto Gorrieri ; Alberto Marchetti-Spaccamela. Springer, 1997. pp. 760-770 (Lecture notes in computer science).
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abstract = "An asteroidal triple is a set of three vertices such that there is a path between any pair of them avoiding the closed neighborhood of the third. A graph is called AT-free if it does not have an asteroidal triple. We show that there is an O(n 2 · (¯m+1)) time algorithm to compute the maximum cardinality of an independent set for AT-free graphs, where n is the number of vertices and ¯m is the number of non edges of the input graph. Furthermore we obtain O(n 2 · (¯m+1)) time algorithms to solve the INDEPENDENT DOMINATING SET and the INDEPENDENT PERFECT DOMINATING SET problem on AT-free graphs. We also show how to adapt these algorithms such that they solve the corresponding problem for graphs with bounded asteroidal number in polynomial time. Finally we observe that the problems CLIQUE and PARTITION INTO CLIQUES remain NP-complete when restricted to AT-free graphs.",
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Broersma, HJ, Kloks, T, Kloks, AJJ, Kratsch, D & Müller, H 1997, Independent sets in asteroidal triple-free graphs. in P Degano, R Gorrieri & A Marchetti-Spaccamela (eds), Automata, Languages and Programming: 24th International Colloquium, ICALP '97 Bologna, Italy, July 7–11, 1997 Proceedings. Lecture notes in computer science, vol. 1256, Springer, pp. 760-770, 24th International Colloquium on Automata, Languages and Programming, ICALP 1997, Bologna, Italy, 7/07/97. https://doi.org/10.1007/3-540-63165-8_229

Independent sets in asteroidal triple-free graphs. / Broersma, Haitze J.; Kloks, Ton; Kloks, A.J.J.; Kratsch, Dieter; Müller, Haiko.

Automata, Languages and Programming: 24th International Colloquium, ICALP '97 Bologna, Italy, July 7–11, 1997 Proceedings. ed. / Pierpaolo Degano; Roberto Gorrieri; Alberto Marchetti-Spaccamela. Springer, 1997. p. 760-770 (Lecture notes in computer science; Vol. 1256).

Research output: Chapter in Book/Report/Conference proceedingConference contributionAcademic

TY - GEN

T1 - Independent sets in asteroidal triple-free graphs

AU - Broersma, Haitze J.

AU - Kloks, Ton

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AU - Kratsch, Dieter

AU - Müller, Haiko

PY - 1997

Y1 - 1997

N2 - An asteroidal triple is a set of three vertices such that there is a path between any pair of them avoiding the closed neighborhood of the third. A graph is called AT-free if it does not have an asteroidal triple. We show that there is an O(n 2 · (¯m+1)) time algorithm to compute the maximum cardinality of an independent set for AT-free graphs, where n is the number of vertices and ¯m is the number of non edges of the input graph. Furthermore we obtain O(n 2 · (¯m+1)) time algorithms to solve the INDEPENDENT DOMINATING SET and the INDEPENDENT PERFECT DOMINATING SET problem on AT-free graphs. We also show how to adapt these algorithms such that they solve the corresponding problem for graphs with bounded asteroidal number in polynomial time. Finally we observe that the problems CLIQUE and PARTITION INTO CLIQUES remain NP-complete when restricted to AT-free graphs.

AB - An asteroidal triple is a set of three vertices such that there is a path between any pair of them avoiding the closed neighborhood of the third. A graph is called AT-free if it does not have an asteroidal triple. We show that there is an O(n 2 · (¯m+1)) time algorithm to compute the maximum cardinality of an independent set for AT-free graphs, where n is the number of vertices and ¯m is the number of non edges of the input graph. Furthermore we obtain O(n 2 · (¯m+1)) time algorithms to solve the INDEPENDENT DOMINATING SET and the INDEPENDENT PERFECT DOMINATING SET problem on AT-free graphs. We also show how to adapt these algorithms such that they solve the corresponding problem for graphs with bounded asteroidal number in polynomial time. Finally we observe that the problems CLIQUE and PARTITION INTO CLIQUES remain NP-complete when restricted to AT-free graphs.

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T3 - Lecture notes in computer science

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Broersma HJ, Kloks T, Kloks AJJ, Kratsch D, Müller H. Independent sets in asteroidal triple-free graphs. In Degano P, Gorrieri R, Marchetti-Spaccamela A, editors, Automata, Languages and Programming: 24th International Colloquium, ICALP '97 Bologna, Italy, July 7–11, 1997 Proceedings. Springer. 1997. p. 760-770. (Lecture notes in computer science). https://doi.org/10.1007/3-540-63165-8_229