Skip to main content
SpringerLink
Log in

Maximal Cliques Lattices Structures for Cocomparability Graphs with Algorithmic Applications

  • Published:
Order Aims and scope Submit manuscript

Abstract

A cocomparability graph is a graph whose complement admits a transitive orientation. An interval graph is the intersection graph of a family of intervals on the real line. In this paper we investigate the relationships between interval and cocomparability graphs. This study is motivated by recent results (Corneil et al., SIAM J. Comput. 42(3):792–807, 2013; Dusart et al., SIAM J. Comput. 50(3):1148–1153, 2021; Dusart and Habib, Discret. Appl. Math. 216:149–161, 2017) that show that for some problems, the algorithm used on interval graphs can also be used with small modifications on cocomparability graphs. Many of these algorithms are based on graph searches that preserve cocomparability orderings. First we propose a characterization of cocomparability graphs via a lattice structure on the set of their maximal cliques. Using this characterization we can prove that every maximal interval subgraph of a cocomparability graph G is also a maximal chordal subgraph of G. Although the size of this lattice of maximal cliques can be exponential in the size of the graph, it can be used as a framework to design and prove algorithms on cocomparability graphs. In particular we show that a new graph search, namely Local Maximal Neighborhood Search (LocalMNS) leads to an algorithm to find in linear time a maximal interval subgraph of a cocomparability graph which improves on the current state of knowledge. Although computing all simplicial vertices is known to be equivalent to the recognition of triangle-free graphs or boolean multiplication, see (Krastch and Spinrad, SIAM J. Comput. 36(2):310–325, 2007; Coudert and Ducoffe, SIAM J. Discrete Math. 32(1):682–694, 2018), we will show that our structural insights in cocomparability graphs together with the definition of a new graph search, allow us to achieve linear time on this class of graphs.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Availability of data and materials

Data sharing not applicable to this article as no datasets were generated or analysed during the current study.

References

  1. Behrendt, B.: Maximal antichains in partially ordered sets. Ars Comb. 25C, 149–157 (1988)

    MathSciNet  Google Scholar 

  2. Birkhoff, G.: Lattice theory. Am. Math. Soc. 25(3), (1967)

  3. Brandstädt, A., Le, V.B., Spinrad, J.P.: Graph Classes, a survey. SIAM Monogr. Discret. Math. Appl. (1999)

  4. Caspard, N., Leclerc, B., Monjardet, B.: Finite Ordered Sets Concepts, Results and Uses. Cambridge University Press (2012)

  5. Corneil, D.G., Dalton, B., Habib, M.: LDFS-based certifying algorithm for the minimum path cover problem on cocomparability graphs. SIAM J. Comput. 42(3), 792–807 (2013)

    Article  MathSciNet  Google Scholar 

  6. Corneil, D.G., Dusart, J., Habib, M., Köhler, E.: On the power of graph searching for cocomparability graphs. SIAM J. Discret. Math. 30(1), 569–591 (2016)

    Article  MathSciNet  Google Scholar 

  7. Corneil, D.G., Krueger, R.: A unified view of graph searching. SIAM J. Disc. Math. 22, 1259–1276 (2008)

    Article  MathSciNet  Google Scholar 

  8. Corneil, D.G., Olariu, S., Stewart, L.: Linear time algorithms for dominating pairs in asteroidal triple-free graphs. SIAM J. Comput. 28(4), 1284–1297 (1999)

    Article  MathSciNet  Google Scholar 

  9. Corneil, D.G., Olariu, S., Stewart, L.: The LBFS structure and recognition of interval graphs. SIAM J. Discret. Math. 23(4), 1905–1953 (2009)

    Article  MathSciNet  Google Scholar 

  10. Coudert, D., Ducoffe, G.: Revisiting Decomposition by Clique Separators. SIAM J. Discret. Math. 32(1), 682–694 (2018)

    Article  MathSciNet  Google Scholar 

  11. Crespelle, C., Todinca, I.: An \({O}(n^2)\)-time algorithm for the minimal interval completion problem. Theor. Comput. Sci. 494, 75–85 (2013)

  12. Davey, B. A., Priestley, H. A.: Introduction to Lattices and Order (2nd ed.). Cambridge University Press, (2002)

  13. Dearing, P.M., Shier, D.R., Warner, D.D.: Maximal chordal subgraphs. Discret. Appl. Math. 20(3), 181–190 (1988)

    Article  MathSciNet  Google Scholar 

  14. Dilworth, R.P.: Some combinatorial problems on partially ordered sets. Proc. Symp. Appl. Math. (Ann. Math. Soc. Providence R.I.) 10, 85–90 (1960)

  15. Dusart, J.: Graph Searches with Applications to Cocomparability Graphs. PhD thesis, Université Paris Diderot, (2014)

  16. Dusart, J., Corneil, D.G., Habib, M.: Corrigendum: LDFS-Based Certifying Algorithm for the Minimum Path Cover Problem on Cocomparability Graphs. SIAM J. Comput. 50(3), 1148–1153 (2021)

    Article  MathSciNet  Google Scholar 

  17. Dusart, J., Habib, M.: A new LBFS-based algorithm for cocomparability graph recognition. Discret. Appl. Math. 216, 149–161 (2017)

    Article  MathSciNet  Google Scholar 

  18. Dushnik, B., Miller, E.W.: Partially ordered sets. Am. J. Math. 63(3), 600–610 (1941)

    Article  MathSciNet  Google Scholar 

  19. Fishburn, P.C.: Interval orders and interval graphs. Wiley, (1985)

  20. Galinier, P., Habib, M., Paul, C.: Chordal Graphs and Their Clique Graphs. WG95, Graph-Theoretic Concepts in Computer Science, 21st International Workshop, LNCS 1017. pp. 358–371 (1995)

  21. Garbe, R.: Algorithmic aspects of interval orders. PhD Thesis, Univ. of Twente, Enschede, The Netherlands (1994)

  22. Gilmore, P.C., Hoffman, A.J.: A characterization of comparability graphs and of interval graphs. Canad. J. Math. 16, 539–548 (1964)

    Article  MathSciNet  Google Scholar 

  23. Golumbic, M.C.: Algorithmic Graph Theory and Perfect Graphs. Annals of Discrete Mathematics, vol. 57. North-Holland Publishing Co., Amsterdam (2004)

    Google Scholar 

  24. Grätzer, G.: General Lattice Theory. Birkhäuser (1968)

  25. 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  MathSciNet  Google Scholar 

  26. Habib, M., Möhring, R.H.: Treewidth of cocomparability graphs and a new order-theoretic parameter. Order 11(3), 47–60 (1991)

    MathSciNet  Google Scholar 

  27. Habib, M., Morvan, M., Pouzet, M., Rampon, J.-X.: Extensions intervallaires minimales. In: Compte Rendu à l’Académie des Sciences Paris, présenté en septembre 91, par le Pr. G. Choquet, vol 313. pp. 893–898 (1991)

  28. Habib, M., Morvan, M., Pouzet, M., Rampon, J.-X.: Incidence structures, coding and lattice of maximal antichains. Technical Report 92-079, LIRMM, (1992)

  29. Heggernes, P., Suchan, K., Todinca, I., Villanger, Y.: Minimal interval completions. In: Brodal, G. S., Leonardi, S. (eds.) ESA. Lecture Notes in Computer Science, vol 3669. pp. 403–414. Springer (2005)

  30. Jakubík, Ján.: Maximal antichains in a partially ordered set. Czechoslov. Math. J. 41(1), 75–84 (1991)

    Article  MathSciNet  Google Scholar 

  31. Köhler, E., Mouatadid, L.: Linear time LexDFS on cocomparability graphs. In: Algorithm Theory - SWAT 2014 - 14th Scandinavian Symposium and Workshops, Copenhagen, Denmark, July 2-4, 2014. Proceedings. pp. 319–330 (2014)

  32. Krastch, D., Spinrad, J.: Between \(O(mn)\) and \(0(n^{\alpha })\). SIAM J. Comput. 36(2), 310–325 (2007)

  33. Kratsch, D., Stewart, L.: Domination on cocomparability graphs. SIAM J. Discret. Math. 6(3), 400–417 (1993)

    Article  MathSciNet  Google Scholar 

  34. Lekkerkerker, C., Boland, J.: Representation of a finite graph by a set of intervals on the real line. Fundam. Math. 51(1), 45–64 (1962)

    Article  MathSciNet  Google Scholar 

  35. Ma, T.-H.: Unpublished manuscript. (1992)

  36. Markowsky, G.: The factorization and representation of lattices. Trans. Am. Math. Soc. 203, 185–200 (1975)

    Article  MathSciNet  Google Scholar 

  37. Markowsky, G.: Primes, irreductibles and extremal lattices. Order 9, 265–290 (1992)

    Article  MathSciNet  Google Scholar 

  38. McConnell, R.M., Spinrad, J.: Modular decomposition and transitive orientation. Discret. Math. 201(1–3), 189–241 (1999)

    Article  MathSciNet  Google Scholar 

  39. Meister, D.: Recognition and computation of minimal triangulations for AT-free claw-free and co-comparability graphs. Discret. Appl. Math. 146(3), 193–218 (2005)

    Article  MathSciNet  Google Scholar 

  40. Mertzios, G.B., Corneil, D.G.: A simple polynomial algorithm for the longest path problem on cocomparability graphs. SIAM J. Discret. Math. 26(3), 940–963 (2012)

    Article  MathSciNet  Google Scholar 

  41. Mertzios, G.B., Nichterlein, A., Niedermeier, R.: A Linear-Time Algorithm for Maximum-Cardinality Matching on Cocomparability Graphs. SIAM J. Discret. Math. 32(4), 2820–2835 (2018)

    Article  MathSciNet  Google Scholar 

  42. Möhring, R.H.: Algorithmic aspects of comparability graphs and interval graphs. In: Rival, I. (ed.) Graphs and Order. NATO ASI Series, vol. 147, pp. 41–101. Springer Netherlands (1985)

  43. Möhring, R.H.: Triangulating graphs without asteroidal triples. Discrete Appl. Math. 64(3), 281–287 (1996)

    Article  MathSciNet  Google Scholar 

  44. Papadimitriou, C.H., Yannakakis, M.: Scheduling interval ordered tasks. SIAM J. Comput. 8, 405–409 (1979)

    Article  MathSciNet  Google Scholar 

  45. Parra, A.: Structural and algorithmic aspects of chordal graph embeddings. PhD thesis, Technische Universität Berlin, 1996

  46. Reuter, K.: The jump number and the lattice of maximal antichains. Discrete Mathematics 88, 289–307 (1991)

    Article  MathSciNet  Google Scholar 

  47. Rose, D.J., Tarjan, R.E., Lueker, G.S.: Algorithmic Aspects of Vertex Elimination on Graphs. SIAM J. Comput. 5, 266–283 (1976)

    Article  MathSciNet  Google Scholar 

  48. Schröder, B. S. W.: Ordered sets, an introduction. Birkhäuser (2002)

  49. Simon, K.: A New Simple Linear Algorithm to Recognize Interval Graphs. Work. Comput. Geom. 289–308 (1991)

  50. Trotter, W.T.: Combinatorial aspects of interval orders and interval graphs. Electron. Notes Discret. Math. 2, 153 (1999)

    Article  Google Scholar 

Download references

Acknowledgements

The authors wish to thank the referee for his very fruitful comments.

The third author wishes to thank the Natural Sciences and Engineering Research Council (NSERC) of Canada for financial support of this research.

Author information

Authors and Affiliations

  1. IRIF, CNRS & Université Paris Cité, Paris, France

    Jérémie Dusart & Michel Habib

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

    Derek G. Corneil

Authors
  1. Jérémie Dusart
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Michel Habib
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Derek G. Corneil
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Michel Habib.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Maurice thanks for all the energy you put into passing on your passion for ordered sets!.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Dusart, J., Habib, M. & Corneil, D.G. Maximal Cliques Lattices Structures for Cocomparability Graphs with Algorithmic Applications. Order 41, 99–133 (2024). https://doi.org/10.1007/s11083-023-09641-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11083-023-09641-x

Keywords

Search

Navigation