Location via proxy:   [ UP ]  
[Report a bug]   [Manage cookies]                
Skip to main content
SpringerLink
Log in

Practical and Efficient Circle Graph Recognition

  • Published:
Algorithmica Aims and scope Submit manuscript

Abstract

Circle graphs are the intersection graphs of chords in a circle. This paper presents the first sub-quadratic recognition algorithm for the class of circle graphs. Our algorithm is O(n+m) times the inverse Ackermann function, α(n+m), whose value is smaller than 4 for any practical graph. The algorithm is based on a new incremental Lexicographic Breadth-First Search characterization of circle graphs, and a new efficient data-structure for circle graphs, both developed in the paper. The algorithm is an extension of a Split Decomposition algorithm with the same running time developed by the authors in a companion paper.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Algorithm 1
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Algorithm 2
Fig. 9

Similar content being viewed by others

Notes

  1. Let us mention that several definitions exist for this function, either with two variables, including some variants, or with one variable. For simplicity, we choose to use the version with one variable. This makes no practical difference since all of them could be used in our complexity bound, and they are all essentially constant. As an example, the two variable function considered in [3] satisfies α(k,n)≤4 for all integer k and for all \(n\leq\underbrace{2^{.^{.^{.^{2}}}}}_{17 \mathrm{\ times}}\).

References

  1. Bouchet, A.: Reducing prime graphs and recognizing circle graphs. Combinatorica 7, 243–254 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  2. Bouchet, A.: Graphic presentations of isotropic systems. J. Comb. Theory, Ser. B 45, 58–76 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  3. Cormen, T.H., Stein, C., Rivest, R.L., Leiserson, C.E.: Introduction to Algorithms. McGraw-Hill, New York (2001)

    MATH  Google Scholar 

  4. Corneil, D.G.: Lexicographic breadth first search—a survey. In: International Workshop on Graph Theoretical Concepts in Computer Science (WG). Lecture Notes in Computer Science, vol. 3353, pp. 1–19 (2004)

    Chapter  Google Scholar 

  5. Courcelle, B.: The monadic second-order logic of graphs XVI: canonical graph decomposition. Log. Methods Comput. Sci. 2(2), 1–46 (2006)

    Article  MathSciNet  Google Scholar 

  6. Courcelle, B.: Circle graphs and monadic second-order logic. J. Appl. Log. 6(3), 416–442 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  7. Cunningham, W.H.: Decomposition of directed graphs. SIAM J. Algebr. Discrete Methods 3, 214–228 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  8. Dahlhaus, E.: Parallel algorithms for hierarchical clustering and applications to split decomposition and parity graph recognition. J. Algorithms 36(2), 205–240 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  9. Dragan, F., Nicolai, F., Brandstädt, A.: LexBFS-orderings and powers of graphs. In: International Workshop on Graph Theoretical Concepts in Computer Science (WG). Lecture Notes in Computer Science, vol. 1197, pp. 166–180 (1996)

    Chapter  Google Scholar 

  10. Even, S., Itai, A.: Queues, stacks and graphs. In: Theory of Machines and Computations, pp. 71–86 (1971)

    Google Scholar 

  11. Gabor, C.P., Hsu, W.L., Suppovit, K.J.: Recognizing circle graphs in polynomial time. J. ACM 36, 435–473 (1989)

    Article  MATH  Google Scholar 

  12. Geelen, J., Oum, S.-I.: Circle graph obstructions under pivoting. J. Graph Theory 61(1), 1–11 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  13. Gioan, E., Paul, C.: Dynamic distance hereditary graphs using split decomposition. In: International Symposium on Algorithms and Computation (ISAAC). Lecture Notes in Computer Science, vol. 4835, pp. 41–51 (2007)

    Chapter  Google Scholar 

  14. Gioan, E., Paul, C.: Split decomposition and graph-labelled trees: characterizations and fully-dynamic algorithms for totally decomposable graphs. Discrete Appl. Math. 160(6), 708–733 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  15. Gioan, E., Paul, C., Tedder, M., Corneil, D.: Practical and efficient split decomposition via graph-labelled trees. Algorithmica (2013). doi:10.1007/s00453-013-9752-9

    Google Scholar 

  16. Golumbic, M.C.: Algorithmic Graph Theory and Perfect Graphs, 2nd edn. Elsevier, Amsterdam (2004)

    MATH  Google Scholar 

  17. Habib, M., McConnell, R.M., Paul, C., Viennot, L.: Lex-BFS and partition refinement, with applications to transitive orientation, interval graph recognition and consecutive ones testing. Theor. Comput. Sci. 234(1–2), 59–84 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  18. Ma, T.-H., Spinrad, J.: An O(n 2) algorithm for undirected split decomposition. J. Algorithms 16, 145–160 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  19. Naji, W.: Reconnaissance des graphes de cordes. Discrete Math. 54, 329–337 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  20. Oum, S.-I.: Rank-width and vertex minors. J. Comb. Theory, Ser. B 95(1), 79–100 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  21. Oum, S.-I.: Excluding a bipartite circle graph from line graphs. J. Graph Theory 60(3), 183–203 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  22. Rose, D.J., Tarjan, R.E., Lueker, G.S.: Algorithmic aspects of vertex elimination on graphs. SIAM J. Comput. 5(2), 266–283 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  23. Spinrad, J.: Recognition of circle graphs. J. Algorithms 16, 264–282 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  24. Tarjan, R.E.: Efficiency of a good but not linear set union algorithm. J. ACM 22(2), 146–160 (1975)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. CNRS, LIRMM, Université Montpellier 2, Montpellier, France

    Emeric Gioan & Christophe Paul

  2. Department of Computer Science, University of Toronto, Toronto, Canada

    Marc Tedder & Derek Corneil

Authors
  1. Emeric Gioan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Christophe Paul
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Marc Tedder
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Derek Corneil
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Christophe Paul.

Additional information

Financial support of E. Gioan and C. Paul was received from the French ANR project ANR-O6-BLAN-0148-01: Graph Decomposition and Algorithms (GRAAL). Financial support of M. Tedder and D. Corneil was received from Canada’s Natural Sciences and Engineering Research Council (NSERC).

Rights and permissions

Reprints and permissions

About this article

Cite this article

Gioan, E., Paul, C., Tedder, M. et al. Practical and Efficient Circle Graph Recognition. Algorithmica 69, 759–788 (2014). https://doi.org/10.1007/s00453-013-9745-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00453-013-9745-8

Keywords

Search

Navigation