Advertisement

Recognizing d-Interval Graphs and d-Track Interval Graphs

  • Minghui Jiang
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6213)

Abstract

A d-interval is the union of d disjoint intervals on the real line. A d-track interval is the union of d disjoint intervals on d disjoint parallel lines called tracks, one interval on each track. As generalizations of the ubiquitous interval graphs, d-interval graphs and d-track interval graphs have wide applications, traditionally to scheduling and resource allocation, and more recently to bioinformatics. In this paper, we prove that recognizing d-track interval graphs is NP-complete for any constant d ≥ 2. This confirms a conjecture of Gyárfás and West in 1995. Previously only the complexity of the d = 2 case was known. Our proof in fact implies that several restricted variants of this graph recognition problem, i.e, recognizing balanced d-track interval graphs, unit d-track interval graphs, and (2,...,2) d-track interval graphs, are all NP-complete. This partially answers another question recently raised by Gambette and Vialette. We also prove that recognizing depth-two 2-track interval graphs is NP-complete, even for the unit case. In sharp contrast, we present a simple linear-time algorithm for recognizing depth-two unit d-interval graphs. These and other results of ours give partial answers to a question of West and Shmoys in 1984 and a similar question of Gyárfás and West in 1995. Finally, we give the first bounds on the track number and the unit track number of a graph in terms of the number of vertices, the number of edges, and the maximum degree, and link the two numbers to the classical concepts of arboricity.

Keywords

Hamiltonian Path Interval Graph Interval Number Complete Bipartite Graph Discrete Apply Mathematic 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Akiyama, J., Exoo, G., Harary, F.: Covering and packing in graphs III: cyclic and acyclic invariants. Mathematica Slovaca 30, 405–417 (1980)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Alcón, L., Cerioli, M.R., de Figueiredo, C.M.H., Gutierrez, M., Meidanis, J.: Tree loop graphs. Discrete Applied Mathematics 155, 686–694 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Alon, N.: The linear arboricity of graphs. Israel Journal of Mathematics 62, 311–325 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Andreae, T.: On the unit interval number of a graph. Discrete Applied Mathematics 22, 1–7 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bafna, V., Narayanan, B., Ravi, R.: Nonoverlapping local alignments (weighted independent sets of axis-parallel rectangles). Discrete Applied Mathematics 71, 41–53 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Balogh, J., Pluhár, A.: A sharp edge bound on the interval number of a graph. Journal of Graph Theory 32, 153–159 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Bar-Yehuda, R., Halldórsson, M.M., Naor, J.(S.), Shachnai, H., Shapira, I.: Scheduling split intervals. SIAM Journal on Computing 36, 1–15 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Butman, A., Hermelin, D., Lewenstein, M., Rawitz, D.: Optimization problems in multiple-interval graphs. In: Proceedings of the 18th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2007), pp. 268–277 (2007)Google Scholar
  9. 9.
    Crochemore, M., Hermelin, D., Landau, G.M., Rawitz, D., Vialette, S.: Approximating the 2-interval pattern problem. Theoretical Computer Science 395, 283–297 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Gambette, P., Vialette, S.: On restrictions of balanced 2-interval graphs. In: Brandstädt, A., Kratsch, D., Müller, H. (eds.) WG 2007. LNCS, vol. 4769, pp. 55–65. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  11. 11.
    Griggs, J.R.: Extremal values of the interval number of a graph, II. Discrete Mathematics 28, 37–47 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Gyárfás, A., West, D.B.: Multitrack interval graphs. Congressus Numerantium 109, 109–116 (1995)MathSciNetzbMATHGoogle Scholar
  14. 14.
    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
  15. 15.
    Jiang, M.: Approximation algorithms for predicting RNA secondary structures with arbitrary pseudoknots. IEEE/ACM Transactions on Computational Biology and Bioinformatics 7, 323–332 (2010)CrossRefGoogle Scholar
  16. 16.
    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)CrossRefGoogle Scholar
  17. 17.
    Kumar, N., Deo, N.: Multidimensional interval graphs. Congressus Numerantium 102, 45–56 (1994)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Maas, C.: Determining the interval number of a triangle-free graph. Computing 31, 347–354 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Roberts, F.S.: Graph Theory and Its Applications to Problems of Society. SIAM, Philadelphia (1987)Google Scholar
  20. 20.
    Trotter Jr., W.T., Harary, F.: On double and multiple interval graphs. Journal of Graph Theory 3, 205–211 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Vialette, S.: On the computational complexity of 2-interval pattern matching problems. Theoretical Computer Science 312, 223–249 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    West, D.B.: A short proof of the degree bound for interval number. Discrete Mathematics 73, 309–310 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    West, D.B., Shmoys, D.B.: Recognizing graphs with fixed interval number is NP-complete. Discrete Applied Mathematics 8, 295–305 (1984)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Minghui Jiang
    • 1
  1. 1.Department of Computer ScienceUtah State UniversityLoganUSA

Personalised recommendations