On Restrictions of Balanced 2-Interval Graphs

  • Philippe Gambette
  • Stéphane Vialette
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4769)


The class of 2-interval graphs has been introduced for modelling scheduling and allocation problems, and more recently for specific bioinformatics problems. Some of those applications imply restrictions on the 2-interval graphs, and justify the introduction of a hierarchy of subclasses of 2-interval graphs that generalize line graphs: balanced 2-interval graphs, unit 2-interval graphs, and (x,x)-interval graphs. We provide instances that show that all inclusions are strict. We extend the NP-completeness proof of recognizing 2-interval graphs to the recognition of balanced 2-interval graphs. Finally we give hints on the complexity of unit 2-interval graphs recognition, by studying relationships with other graph classes: proper circular-arc, quasi-line graphs, K 1,5-free graphs, ...


2-interval graphs graph classes line graphs quasi-line graphs claw-free graphs circular interval graphs bioinformatics scheduling 


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  1. [Ben81]
    Rebea, A.B.: Étude des stables dans les graphes quasi-adjoints. PhD thesis, Université de Grenoble (1981)Google Scholar
  2. [BFV04]
    Blin, G., Fertin, G., Vialette, S.: New results for the 2-interval pattern problem. In: Sahinalp, S.C., Muthukrishnan, S.M., Dogrusoz, U. (eds.) CPM 2004. LNCS, vol. 3109, Springer, Heidelberg (2004)Google Scholar
  3. [BHLR07]
    Butman, A., Hermelin, D., Lewenstein, M., Rawitz, D.: Optimization problems in multiple-interval graphs. In: Proceedings of SODA 2007, pp. 268–277 (2007)Google Scholar
  4. [BLS+]
    Brandstädt, A., Le, V.B., Szymczak, T., Siegemund, F., de Ridder, H.N., Knorr, S., Rzehak, M., Mowitz, M., Ryabova, N.: ISGCI: Information System on Graph Class Inclusions.
  5. [BNR96]
    Bafna, V., Narayanan, B.O., Ravi, R.: Nonoverlapping local alignments (weighted independent sets of axis-parallel rectangles). Discrete Applied Math. 71(1), 41–54 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  6. [BYHN+06]
    Bar-Yehuda, R., Halldórson, M.M., Naor, J., Shachnai, H., Shapira, I.: Scheduling split intervals. SIAM Journal on Computing 36(1), 1–15 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  7. [CHLV05]
    Crochemore, M., Hermelin, D., Landau, G.M., Vialette, S.: Approximating the 2-interval pattern problem. In: Brodal, G.S., Leonardi, S. (eds.) ESA 2005. LNCS, vol. 3669, pp. 426–437. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  8. [CKN+95]
    Corneil, D.G., Kim, H., Natarajan, S., Olariu, S., Sprague, A.P.: Simple linear time recognition of unit interval graphs. Information Processing Letters 55, 99–104 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  9. [CS05]
    Chudnovsky, M., Seymour, P.: The structure of claw-free graphs. In: Surveys in Combinatorics. London. Math. Soc. Lecture Notes, vol. 327, pp. 153–172. Cambridge University Press, Cambridge (2005)Google Scholar
  10. [FFR97]
    Faudree, R., Flandrin, E., Ryjáček, Z.: Claw-free graphs - a survey. Discrete Mathematics 164, 87–147 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  11. [GW80]
    Griggs, J.R., West, D.B.: Extremal values of the interval number of a graph. SIAM Journal on Algebraic and Discrete Methods 1, 1–7 (1980)zbMATHMathSciNetGoogle Scholar
  12. [GW95]
    Gyárfás, A., West, D.B.: Multitrack interval graphs. Congress Numerantium 109, 109–116 (1995)zbMATHGoogle Scholar
  13. [HK06]
    Halldórsson, M.M., Karlsson, R.K.: Strip graphs: Recognition and scheduling. In: Fomin, F.V. (ed.) WG 2006. LNCS, vol. 4271, pp. 137–146. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  14. [JMT92]
    Joseph, D., Meidanis, J., Tiwari, P.: Determining DNA sequence similarity using maximum independent set algorithms for interval graphs. In: Nurmi, O., Ukkonen, E. (eds.) SWAT 1992. LNCS, vol. 621, pp. 326–337. Springer, Heidelberg (1992)Google Scholar
  15. [Kar72]
    Karp, R.M.: Reducibility among combinatorial problems, pp. 85–103. Plenum Press, New York (1972)Google Scholar
  16. [Kar05]
    Karlsson, R.: A survey of split intervals and related graphs, Manuscript (2005)Google Scholar
  17. [KR07]
    King, A., Reed, B.: Bounding χ in terms of ω ans δ for quasi-line graphs. Article in preparation (2007)Google Scholar
  18. [KW99]
    Kostochka, A.V., West, D.B.: Every outerplanar graph is the union of two interval graphs. Congress Numerantium 139, 5–8 (1999)zbMATHMathSciNetGoogle Scholar
  19. [Lov78]
    Lovász, L.: Kneser’s conjecture, chromatic number, and homotopy. Journal of Combinatorial Theory Series A 25, 319–324 (1978)zbMATHCrossRefMathSciNetGoogle Scholar
  20. [McG77]
    McGuigan, R.: Presentation at NSF-CBMS Conference at Colby College (1977)Google Scholar
  21. [Rob69]
    Roberts, F.S.: Indifference graphs. In: Proof Techniques in Graph Theory, Proceedings of the Second Ann Arbor Graph Theory Conference, pp. 139–146 (1969)Google Scholar
  22. [Via01]
    Vialette, S.: Aspects algorithmiques de la prédiction des structures secondaires d’ARN. PhD thesis, Université Paris 7 (2001)Google Scholar
  23. [Via04]
    Vialette, S.: On the computational complexity of 2-interval pattern matching. Theoretical Computer Science 312(2-3), 223–249 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  24. [WS84]
    West, D.B., Shmoys, D.B.: Recognizing graphs with fixed interval number is NP-complete. Discrete Applied Math. 8, 295–305 (1984)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Philippe Gambette
    • 1
  • Stéphane Vialette
    • 2
  1. 1.LIAFA - Univ. Paris VII 
  2. 2.LRI - Univ. Paris XI 

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