Document Open Access Logo

Linear Time LexDFS on Chordal Graphs

Authors Jesse Beisegel, Ekkehard Köhler, Robert Scheffler, Martin Strehler

PDF
Thumbnail PDF

File

LIPIcs.ESA.2020.13.pdf
  • Filesize: 497 kB
  • 13 pages

Document Identifiers

Author Details

Jesse Beisegel
  • Brandenburg University of Technology, Cottbus, Germany
Ekkehard Köhler
  • Brandenburg University of Technology, Cottbus, Germany
Robert Scheffler
  • Brandenburg University of Technology, Cottbus, Germany
Martin Strehler
  • Brandenburg University of Technology, Cottbus, Germany

Acknowledgements

The authors would like to thank one of the anonymous referees for his or her many helpful comments.

Cite AsGet BibTex

Jesse Beisegel, Ekkehard Köhler, Robert Scheffler, and Martin Strehler. Linear Time LexDFS on Chordal Graphs. In 28th Annual European Symposium on Algorithms (ESA 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 173, pp. 13:1-13:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
https://doi.org/10.4230/LIPIcs.ESA.2020.13

Abstract

Lexicographic Depth First Search (LexDFS) is a special variant of a Depth First Search (DFS), which was introduced by Corneil and Krueger in 2008. While this search has been used in various applications, in contrast to other graph searches, no general linear time implementation is known to date. In 2014, Köhler and Mouatadid achieved linear running time to compute some special LexDFS orderings for cocomparability graphs. In this paper, we present a linear time implementation of LexDFS for chordal graphs. Our algorithm even implements the extended version LexDFS^+ and is, therefore, able to find any LexDFS ordering for this graph class. To the best of our knowledge this is the first unrestricted linear time implementation of LexDFS on a non-trivial graph class. In the algorithm we use a search tree computed by Lexicographic Breadth First Search (LexBFS).

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Graph algorithms
  • Mathematics of computing → Trees
  • Theory of computation → Graph algorithms analysis
Keywords
  • LexDFS
  • chordal graphs
  • linear time implementation
  • search trees
  • LexBFS

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

References

  1. Jesse Beisegel, Carolin Denkert, Ekkehard Köhler, Matjaž Krnc, Nevena Pivač, Robert Scheffler, and Martin Strehler. The recognition problem of graph search trees. Submitted. Google Scholar
  2. Jesse Beisegel, Carolin Denkert, Ekkehard Köhler, Matjaž Krnc, Nevena Pivač, Robert Scheffler, and Martin Strehler. Recognizing graph search trees. In Proceedings of Lagos 2019, the tenth Latin and American Algorithms, Graphs and Optimization Symposium, volume 346 of ENTCS, pages 99-110. Elsevier, 2019. URL: https://doi.org/10.1016/j.entcs.2019.08.010.
  3. Anne Berry, Richard Krueger, and Geneviève Simonet. Maximal label search algorithms to compute perfect and minimal elimination orderings. SIAM Journal on Discrete Mathematics, 23(1):428-446, 2009. URL: https://doi.org/10.1137/070684355.
  4. Andreas Brandstädt, Van Bang Le, and Jeremy P. Spinrad. Graph Classes: A Survey. SIAM, 1999. URL: https://doi.org/10.1137/1.9780898719796.
  5. Derek G. Corneil, Barnaby Dalton, and Michel Habib. LDFS-based certifying algorithm for the minimum path cover problem on cocomparability graphs. SIAM Journal on Computing, 42(3):792-807, 2013. URL: https://doi.org/10.1137/11083856X.
  6. Derek G. Corneil, Jérémie Dusart, Michel Habib, and Ekkehard Köhler. On the power of graph searching for cocomparability graphs. SIAM Journal on Discrete Mathematics, 30(1):569-591, 2016. URL: https://doi.org/10.1137/15M1012396.
  7. Derek G. Corneil, Jérémie Dusart, Michel Habib, Antoine Mamcarz, and Fabien De Montgolfier. A tie-break model for graph search. Discrete Applied Mathematics, 199:89-100, 2016. URL: https://doi.org/10.1016/j.dam.2015.06.011.
  8. Derek G. Corneil and Richard M. Krueger. A unified view of graph searching. SIAM Journal on Discrete Mathematics, 22(4):1259-1276, 2008. URL: https://doi.org/10.1137/050623498.
  9. Derek G. Corneil, Stephan Olariu, and Lorna Stewart. The LBFS structure and recognition of interval graphs. SIAM Journal on Discrete Mathematics, 23(4):1905-1953, 2009. URL: https://doi.org/10.1137/S0895480100373455.
  10. Jean Creusefond, Thomas Largillier, and Sylvain Peyronnet. A LexDFS-based approach on finding compact communities. In Mehmet Kaya, Özcan Erdoǧan, and Jon Rokne, editors, From Social Data Mining and Analysis to Prediction and Community Detection, pages 141-177. Springer, Cham, 2017. URL: https://doi.org/10.1007/978-3-319-51367-6_7.
  11. Michel Habib, Ross McConnell, Christophe Paul, and Laurent Viennot. Lex-BFS and partition refinement, with applications to transitive orientation, interval graph recognition and consecutive ones testing. Theoretical Computer Science, 234(1-2):59-84, 2000. URL: https://doi.org/10.1016/S0304-3975(97)00241-7.
  12. Torben Hagerup. Biconnected graph assembly and recognition of DFS trees. Technical Report A 85/03, Universität des Saarlandes, 1985. URL: https://doi.org/10.22028/D291-26437.
  13. Torben Hagerup and Manfred Nowak. Recognition of spanning trees defined by graph searches. Technical Report A 85/08, Universität des Saarlandes, 1985. Google Scholar
  14. John Hopcroft and Robert Tarjan. Efficient planarity testing. Journal of the ACM, 21:549-568, 1974. URL: https://doi.org/10.1145/321850.321852.
  15. Ekkehard Köhler and Lalla Mouatadid. Linear time LexDFS on cocomparability graphs. In R. Ravi and Inge Li Gørtz, editors, Algorithm Theory – SWAT 2014, volume 8503 of LNCS, pages 319-330, Cham, 2014. Springer. URL: https://doi.org/10.1007/978-3-319-08404-6_28.
  16. Ephraim Korach and Zvi Ostfeld. DFS tree construction: Algorithms and characterizations. In Jan van Leeuwen, editor, Graph-Theoretic Concepts in Computer Science, volume 344 of LNCS, pages 87-106, Berlin, Heidelberg, 1989. Springer. URL: https://doi.org/10.1007/3-540-50728-0_37.
  17. Richard M. Krueger. Graph Searching. PhD thesis, University of Toronto, 2005. URL: http://www.cs.toronto.edu/~krueger/papers/thesis.ps.
  18. Tze-Heng Ma. Unpublished manuscript. Google Scholar
  19. Udi Manber. Recognizing breadth-first search trees in linear time. Information Processing Letters, 34(4):167-171, 1990. URL: https://doi.org/10.1016/0020-0190(90)90155-Q.
  20. George B. Mertzios, André Nichterlein, and Rolf Niedermeier. A linear-time algorithm for maximum-cardinality matching on cocomparability graphs. SIAM Journal on Discrete Mathematics, 32(4):2820-2835, 2018. URL: https://doi.org/10.1137/17M1120920.
  21. Donald J. Rose, R. Endre Tarjan, and George S. Lueker. Algorithmic aspects of vertex elimination on graphs. SIAM Journal on Computing, 5(2):266-283, 1976. URL: https://doi.org/10.1137/0205021.
  22. Klaus Simon. A new simple linear algorithm to recognize interval graphs. In Hanspeter Bieri and Hartmut Noltemeier, editors, Computational Geometry - Methods, Algorithms and Applications, volume 553 of LNCS, pages 289-308, Berlin, Heidelberg, 1991. Springer. URL: https://doi.org/10.1007/3-540-54891-2_22.
  23. Jeremy P. Spinrad. Efficient implementation of lexicographic depth first search. Submitted. Google Scholar
  24. Robert Tarjan. Depth-first search and linear graph algorithms. SIAM Journal on Computing, 1(2):146-160, 1972. URL: https://doi.org/10.1137/0201010.
  25. Shou-Jun Xu, Xianyue Li, and Ronghua Liang. Moplex orderings generated by the LexDFS algorithm. Discrete Applied Mathematics, 161(13-14):2189-2195, 2013. URL: https://doi.org/10.1016/j.dam.2013.02.028.
© 2023-2024 Schloss Dagstuhl – LZI GmbH Imprint Privacy Contact