Skip to main content

Computational aspects of the Helly property: a survey

Abstract

In 1923, Eduard Helly published his celebrated theorem, which originated the well known Helly property. Say that a family of subsets has the Helly property when every subfamily of it, formed by pairwise intersecting subsets, contains a common element. There are many generalizations of this property which are relevant to some parts of mathematics and several applications in computer science. In this work, we survey computational aspects of the Helly property. The main focus is algorithmic. That is, we describe algorithms for solving different problems arising from the basic Helly property. We also discuss the complexity of these problems, some of them leading to NP-hardness results.

References

  1. [1]

    M. O. Albertson and K. L. Collins. Duality and perfection for edges in cliques.Journal of Combinatorial Theory, Series B, 36:298–309, 1984.

    MATH  Article  MathSciNet  Google Scholar 

  2. [2]

    L. Alcón, L. Faria, C. M. H. Figueiredo, and M. Gutierrez. Clique graphs is NP-complete.Manuscript, 2006.

  3. [3]

    L. Alcón and M. Gutierrez. Cliques and extended triangles. A necessary condition for planar clique graphs.Discrete Applied Mathematics, 141:3–17, 2004.

    MATH  Article  MathSciNet  Google Scholar 

  4. [4]

    N. Alon and D. Kleitman. Piercing convex sets and the Hadwiger-Debrunner (p,q)-problem.Advances in Mathematics, 96:103–112, 1992.

    MATH  Article  MathSciNet  Google Scholar 

  5. [5]

    N. Amenta. Helly theorems and generalized linear programming. InSymposium on Computational Geometry, pages 63-72, 1993.

  6. [6]

    H-J Bandelt, M. Farber, and P. Hell. Absolute reflexive retracts and absolute bipartite retracts.Discrete Applied Mathematics, 44(1–3):9–20, 1993.

    MATH  Article  MathSciNet  Google Scholar 

  7. [7]

    H. J. Bandelt and E. Pesch. Dismantling absolute retracts of reflexive graphs.European Journal of Combinatorics, 10:210–220, 1989.

    MathSciNet  Google Scholar 

  8. [8]

    H-J Bandelt and E. Pesch. Efficient characterizations of n-chromatic absolute retracts.Journal on Combinatorial Theory Series B, 53:5–31, 1991.

    MATH  Article  MathSciNet  Google Scholar 

  9. [9]

    A. Barg, G. Cohen, S. Encheva, G. Kabatiansky, and G. Zémor. A hypergraph approach to the identifying parent property: The case of multiple parents.SIAM Journal on Discrete Mathematics, 14(3):423–431, 2001.

    MATH  Article  MathSciNet  Google Scholar 

  10. [10]

    M. Benke. Efficient type reconstruction in the presence of inheritance. In A. M. Borzyszkowski and S. Sokolowski, editors,Proceedings of Mathematical Foundations of Computer Science (MFCS ’93), volume 711 ofLNCS, pages 272–280, Berlin, Germany, 1993. Springer.

    Google Scholar 

  11. [11]

    C. Berge.Graphes et Hypergraphes. Dunod, Paris, 1970. (Graphs and Hypergraphs, North-Holland, Amsterdam, 1973, revised translation).

    MATH  Google Scholar 

  12. [12]

    C. Berge.Hypergraphs. Gauthier-Villars, Paris, 1987.

    MATH  Google Scholar 

  13. [13]

    C. Berge and P. Duchet. A generalization of Gilmore’s theorem. In M. Fiedler, editor,Recent Advances in Graph Theory, pages 49–55. Acad. Praha, Prague, 1975.

    Google Scholar 

  14. [14]

    B. Bollobás.Combinatorics. Cambridge University Press, Cambridge, 1986.

    MATH  Google Scholar 

  15. [15]

    A. Bondy, G. Durán, M. C. Lin, and J. L. Szwarcfiter. Self-clique graphs and matrix permutations.J. Graph Theory, 44(3):178–192, 2003.

    MATH  Article  MathSciNet  Google Scholar 

  16. [16]

    F. Bonomo. Self-clique Helly circular-arc graphs.Discrete Mathematics, 306:595–597, 2006.

    MATH  Article  MathSciNet  Google Scholar 

  17. [17]

    F. Bonomo, M. Chudnovski, and G. Durán. Partial characterizations of clique-perfect graphs.Eletronic Notes on Discrete Mathematics, 19:95–101, 2005.

    Article  Google Scholar 

  18. [18]

    C. F. Bornstein and J. L. Szwarcfiter. On clique convergent graphs.Graphs and Combinatorics, 11:213–220, 1995.

    MATH  Article  MathSciNet  Google Scholar 

  19. [19]

    A. Brandstädt, V. Chepoi, and F. Dragan. The algorithmic use of hypertree structure and maximum neighbourhood orderings.Discrete Applied Mathematics, 82(1–3):43–77, 1998.

    MATH  Article  MathSciNet  Google Scholar 

  20. [20]

    A. Brandstädt, V. Chepoi, F. Dragan, and V. Voloshin. Dually chordal graphs.SIAM Journal on Discrete Mathematics, 11(3):437–455, aug 1998.

    MATH  Article  MathSciNet  Google Scholar 

  21. [21]

    A. Brandstädt, V. B. Le, and J. P. Spinrad.Graph classes: A survey. SIAM Monographs on Discrete Mathematics and Applications. Society for Industrial and Applied Mathematics, Philadelphia, PA, 1999.

    Google Scholar 

  22. [22]

    A. Bretto, J. Azema, H. Cherifi, and B. Laget. Combinatorics and image processing.Grafical Models and Image Processing, 59(5):265–277, 1997.

    Article  Google Scholar 

  23. [23]

    A. Bretto, S. Ubéda, and J. Žerovnik. A polynomial algorithm for the strong Helly property.Information Processing Letters, 81:55–57, 2002.

    MATH  Article  MathSciNet  Google Scholar 

  24. [24]

    P. L. Butzer, R. J. Nessel, and E. L. Stark.Eduard Helly (1884–1943): In memoriam, volume 7. Resultate der Mathematik, 1984.

  25. [25]

    M. R. Cerioli.Edge-clique Graphs (in portuguese). Ph.D. Thesis, COPPE - Sistemas, Universidade Federal do Rio de Janeiro, Rio de Janeiro, 1999.

    Google Scholar 

  26. [26]

    S. A. Cook. The complexity of theorem-proving procedures.Proc. 3rd Ann. ACM Symp. on Theory of Computing Machinery, New York, pages 151-158, 1971.

  27. [27]

    L. Danzer, B. Grünbaum, and V. L. Klee. Helly’s theorem and its relatives. InProc. Symp. on Pure Math AMS, volume 7, pages 101-180, 1963.

  28. [28]

    M. C. Dourado, P. Petito, and R. B. Teixeira. Helly property and sandwich graphs. InProceedings of ICGT 2005, volume 22 ofEletronic Notes in Discrete Mathematics, pages 497-500. Elsevier B.V., 2005.

  29. [29]

    M. C. Dourado, F. Protti, and J. L. Szwarcfiter. On the Helly defect of a graph.Journal of the Brazilian Computer Society, 7(3):48–52, 2001.

    Article  Google Scholar 

  30. [30]

    M. C. Dourado, F. Protti, and J. L. Szwarcfiter. Characterization and recognition of generalized clique-Helly graphs. In J. Hromkovič, M. Nagl, and B. Westfechtel, editors,Proceedings WG 2004, volume 3353 ofLecture Notes in Computer Science, pages 344-354. Springer-Verlag, 2004.

  31. [31]

    M. C. Dourado, F. Protti, and J. L. Szwarcfiter. The Helly property on subfamilies of limited size.Information Processing Letters, 93:53–56, 2005.

    MATH  MathSciNet  Google Scholar 

  32. [32]

    M. C. Dourado, F. Protti, and J. L. Szwarcfiter. On Helly hypergraphs with predescribed intersection sizes. Submitted, 2005.

  33. [33]

    M. C. Dourado, F. Protti, and J. L. Szwarcfiter. Complexity aspects of generalized Helly hypergraphs.Information Processing Letters, 99:13-18, 2006.

    Google Scholar 

  34. [34]

    M. C. Dourado, F. Protti, and J. L. Szwarcfiter. On the strongp-Helly property.Discrete Applied Mathematics, to appear, 2006.

  35. [35]

    F. F. Dragan.Centers of Graphs and the Helly Property (in russian). Ph. D. Thesis, Moldava State University, Chisinău, 1989.

    Google Scholar 

  36. [36]

    F. F. Dragan, C. F. Prisacaru, and V. D. Chepoi. Location problems in graphs and the Helly property.Diskretnája Matematica, 1992.

  37. [37]

    P. Duchet. Proprieté de Helly et problèmes de représentations. InColloquium International CNRS 260, Problémes Combinatoires et Théorie de Graphs, pages 117-118, Orsay, France, 1976. CNRS.

  38. [38]

    P. Duchet. Hypergraphs. In R. L. Graham, M. Grötschel, and L. Lovász, editors,Handbook of Combinatorics, volume 1, pages 381-432, Amsterdam-New York-Oxford, 1995. Elsevier North-Holland.

  39. [39]

    P. Duchet and H. Meyniel. Ensembles convexes dans les graphes. I: Théorèmes de Helly et de Radon pour graphes et surfaces.European Journal of Combinatorics, 4:127–132, 1983.

    MATH  MathSciNet  Google Scholar 

  40. [40]

    G. Durán. Some new results on circle graphs.Matemática Contemporânea, 25:91–106, 2003.

    MATH  Google Scholar 

  41. [41]

    G. Durán, A. Gravano, M. Groshaus, F. Protti, and J. L. Szwarcfiter. On a conjecture concerning Helly circle graphs.Pesquisa Operacional, pages 221-229, 2003.

  42. [42]

    G. Duran and M. C. Lin. Clique graphs of helly circular arc graphs.Ars Combinatoria, 60:255–271, 2001.

    MATH  MathSciNet  Google Scholar 

  43. [43]

    J. Eckhoff. Helly, Radon, and Carathéodory type theorems. InHandbook of Convex Geometry, pages 389-448. North-Holland, 1993.

  44. [44]

    F. Escalante. Ü ber iterierte clique-graphen.Abhandlungender Mathematischen Seminar der Universität Hamburg, 39:59–68, 1973.

    MathSciNet  Google Scholar 

  45. [45]

    R. Fagin. Acyclic database schemes of various degrees: A painless introduction. In G. Ausiello and M. Protasi, editors,Proceedings of the 8th Colloquium on Trees in Algebra and Programming (CAAP’83), volume 159 of LNCS, pages 65-89, L’Aquila, Italy, mar 1983. Springer.

  46. [46]

    R. Fagin. Degrees of acyclicity for hypergraphs and relational database systems.Journal of the Association for Computing Machinery, 30:514–550, 1983.

    MATH  MathSciNet  Google Scholar 

  47. [47]

    C. Flament. Hypergraphes arborés.Discrete Mathematics, 21:223–226, 1978.

    Article  MathSciNet  Google Scholar 

  48. [48]

    F. Gavril. Algorithms on circular-arc graphs.Networks, 4:357–369, 1974.

    MATH  Article  MathSciNet  Google Scholar 

  49. [49]

    M. C. Golumbic and R. E. Jamison. The edge intersection graphs of paths in a tree.Journal of Combinatorial Theory, Series B, 38:8–22, 1985.

    MATH  Article  MathSciNet  Google Scholar 

  50. [50]

    M. C. Golumbic, H. Kaplan, and R. Shamir. Graph sandwich problems.Journal of Algorithms, 19:449–473, 1995.

    MATH  Article  MathSciNet  Google Scholar 

  51. [51]

    J. E. Goodman, R. Pollack, and R. Wenger. Geometric transversal theory. In J. Pach, editor,New Trends in Discrete and Computational Geometry, number 163–198. Springer-Verlag, Berlin, 1993.

    Google Scholar 

  52. [52]

    M. Groshaus and J. L. Szwarcfiter. Hereditary Helly classes of graphs. Submitted, 2005.

  53. [53]

    M. Groshaus and J. L. Szwarcfiter. Biclique-Helly graphs. Submitted, 2006.

  54. [54]

    M. Groshaus and J. L. Szwarcfiter. The biclique matrix of a graph. In preparation, 2006.

  55. [55]

    R. C. Hamelink. A partial characterization of clique graphs.Journal of Combinatorial Theory, 5:192–197, 1968.

    MATH  Article  MathSciNet  Google Scholar 

  56. [56]

    P. Hell.Rétractions de graphes. Ph.D. Thesis, Université de Montreal, 1972.

  57. [57]

    E. Helly. Ueber mengen konvexer koerper mit gemeinschaftlichen punkter, Jahresber.Math. Verein., 32:175–176, 1923.

    MATH  Google Scholar 

  58. [58]

    R. E. Jamison. Partition numbers for trees and ordered sets.Pacific Journal of Mathematics, 96:115–140, 1981.

    MATH  MathSciNet  Google Scholar 

  59. [59]

    F. Larrión, C. P. de Mello, A. Morgana, V. Neumann-Lara, and M. A. Pizaña. The clique operator on cographs and serial graphs.Discrete Mathematics, 282(1–3):183–191, 2004.

    MATH  Article  MathSciNet  Google Scholar 

  60. [60]

    F. Larrión and V. Neumann-Lara. A family of clique divergent graphs with linear growth.Graphs and Combinatorics, 13:263–266, 1997.

    MATH  MathSciNet  Google Scholar 

  61. [61]

    F. Larrión, V. Neumann-Lara, and M. A. Pizaña. Clique divergent clockwork graphs and partial orders.Discrete Applied Mathematics, 141(1–3):195–207, 2004.

    MATH  Article  MathSciNet  Google Scholar 

  62. [62]

    F. Larrión, V. Neumann-Lara, and M. A. Pizaña. On expansive graphs.Manuscript, 2005.

  63. [63]

    F. Larrión, V. Neumann-Lara, and M. A. Pizaña. Graph relations, clique divergence and surface triangulations.Journal of Graph Theory, 51:110–122, 2006.

    MATH  Article  MathSciNet  Google Scholar 

  64. [64]

    F. Larrión, V. Neumann-Lara, M. A. Pizaña, and T. D. Porter. A hierarchy of self-clique graphs.Discrete Mathematics, 282(1–3):193–208, 2004.

    MATH  Article  MathSciNet  Google Scholar 

  65. [65]

    M. C. Lin and J. L. Szwarcfiter. Characterizations and linear time recognition for Helly circular-arc graphs. In12th Annual International Computing and Combinatorics Conference, Lecture Notes in Computer Science, to appear, Taipei, Taiwan, 2006.

  66. [66]

    L. Lovász. Normal hypergraphs and the perfect graph conjecture.Discrete Mathematics, 2:253–267, 1972.

    MATH  Article  MathSciNet  Google Scholar 

  67. [67]

    L. Lovász.Combinatorial Problems and Exercises. North-Holland, Amsterdam, 1979.

    MATH  Google Scholar 

  68. [68]

    L. Lovász. Perfect graphs. In R. W. Beineke and R. J. Wilson, editors,Selected Topics in Graph Theory, pages 55–87. Academic Press, New York, N. Y., 1983.

    Google Scholar 

  69. [69]

    C. L. Lucchesi, C. P. Mello, and J. L. Szwarcfiter. On clique-complete graphs.Discrete Mathematics, 183:247–254, 1998.

    MATH  Article  MathSciNet  Google Scholar 

  70. [70]

    T. A. Mckee and F. R. McMorris.Topics in Intersection Graph Theory. SIAM Monographs on Discrete Mathematics and Applications, Philadelphia, PA, 1999.

    MATH  Book  Google Scholar 

  71. [71]

    J. W. Moon and L. Moser. On cliques in graphs.Israel Journal of Mathematics, 3:23–28, 1965.

    MATH  Article  MathSciNet  Google Scholar 

  72. [72]

    V. Neumman-Lara. A theory of expansive graphs.Manuscript, 1995.

  73. [73]

    T. Nishizeki and N. Chiba.Planar Graphs: Theory and Algorithms. Annals of Discrete Mathematics 32, North Holland, Amsterdam, New York, Oxford, Tokyo, 1988.

    MATH  Google Scholar 

  74. [74]

    M. C. Paul and S. H. Unger. Minimizing the number of states in incompletely specified sequential switching functions.IRE Transactions Eletronic Computers EC-8, pages 356-367, 1959.

  75. [75]

    M. A. Pizaña. The icosahedron is clique divergent.Discrete Mathematics, 262(1–3):229–239, 2003.

    MATH  Article  MathSciNet  Google Scholar 

  76. [76]

    E. Prisner. Hereditary clique-Helly graphs.Journal of Combinatorial Mathematics and Combinatorial Computing, 14:216–220, 1993.

    MATH  MathSciNet  Google Scholar 

  77. [77]

    E. Prisner.Graph Dynamics. Pitman Research Notes in Mathematics, Longman, 1995.

  78. [78]

    E. Prisner. Bicliques in graphs I: Bounds on their number.Combinatorica, 20(1):109–117, 2000.

    MATH  Article  MathSciNet  Google Scholar 

  79. [79]

    T. M. Przytycka, G. Davis, N. Song, and D. Durand. Graph theoretical insights into evolution of multidomain proteins. InRECOMB 2005 (LNBI, vol. 3500), pages 311-325, 2005.

  80. [80]

    F. S. Roberts and J. H. Spencer. A characterization of clique graphs.Journal of Combinatorial Theory, Series B, 10:102–108, 1971.

    MATH  Article  MathSciNet  Google Scholar 

  81. [81]

    P. J. Slater. A characterization of SOFT hypergraphs.Canadian Mathematical Bulletin, 21:335–337, 1978.

    MATH  Article  MathSciNet  Google Scholar 

  82. [82]

    J. P. Spinrad.Efficient Graph Representation. American Mathematics Society, Providence, RI, 2003.

    Google Scholar 

  83. [83]

    J. L. Szwarcfiter. Recognizing clique-Helly graphs.Ars Combinatoria, 45:29–32, 1997.

    MATH  MathSciNet  Google Scholar 

  84. [84]

    J. L. Szwarcfiter. A survey on clique graphs. In B. A. Reed and C. L. Sales, editors,Recent Advances in Algorithms and Combinatorics, pages 109–136. Springer-Verlag, New York, N. Y., 2003.

    Chapter  Google Scholar 

  85. [85]

    J. L. Szwarcfiter and C. F. Bornstein. Clique graphs of chordal and path graphs.SIAM Journal on Discrete Mathematics, 7(2):331–336, may 1994.

    MATH  Article  MathSciNet  Google Scholar 

  86. [86]

    S. Tsukiyama, M. Ide, H. Ariyoshi, and I. Shirakawa. A new algorithm for generating all the maximal independent sets.SIAM Journal on Computing, 6(3):505–517, sep 1977.

    MATH  Article  MathSciNet  Google Scholar 

  87. [87]

    Zs. Tuza.Extremal Problems on Graphs and Hypergraphs. PhD Thesis, Acad. Sci., Budapeste, 1983. [Hungarian].

    Google Scholar 

  88. [88]

    Zs. Tuza. Helly-type hypergraphs and Sperner families.Europ. J. Combinatorics, 5:185–187, 1984.

    MATH  MathSciNet  Google Scholar 

  89. [89]

    Zs. Tuza. Helly property in finite set systems.Journal of Combinatorial Theory, Series A, 62:1–14, 1993.

    Article  Google Scholar 

  90. [90]

    Zs. Tuza. Extremal bi-Helly families.Discrete Mathematics, 213:321–331, 2000.

    MATH  Article  MathSciNet  Google Scholar 

  91. [91]

    V. I. Voloshin. On the upper chromatic number of a hypergraph.Australasian Journal of Combinatorics, 11:25–45, 1995.

    MATH  MathSciNet  Google Scholar 

  92. [92]

    W. D. Wallis and G.-H. Zhang. On maximal clique irreducible graphs.Journal of Combinatorial Mathematics and Combinatorial Computing, 8:187–193, 1990.

    MATH  MathSciNet  Google Scholar 

Download references

Author information

Affiliations

Authors

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License ( https://creativecommons.org/licenses/by/2.0 ), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Reprints and Permissions

About this article

Cite this article

Dourado, M.C., Protti, F. & Szwarcfiter, J.L. Computational aspects of the Helly property: a survey. J Braz Comp Soc 12, 7–33 (2006). https://doi.org/10.1007/BF03192385

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF03192385

Keywords

  • Computational Complexity
  • Helly property
  • NP-complete problems