A Polynomial Time Algorithm for Longest Paths in Biconvex Graphs

  • Esha Ghosh
  • N. S. Narayanaswamy
  • C. Pandu Rangan
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6552)


The longest path problem is the problem of finding a simple path of maximum length in a graph. Polynomial solutions for this problem are known only for special classes of graphs, while it is NP-hard on general graphs. In this paper we are proposing a O(n 6) time algorithm to find the longest path on biconvex graphs, where n is the number of vertices of the input graph. We have used Dynamic Programming approach.


Longest path problem biconvex graphs polynomial algorithm complexity dynamic programming 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Abbas, N., Stewart, L.K.: Biconvex graphs: ordering and algorithms. Discrete Applied Mathematics 103(1-3), 1–19 (2000)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Rao Arikati, S., Pandu Rangan, C.: Linear algorithm for optimal path cover problem on interval graphs. Inf. Process. Lett. 35(3), 149–153 (1990)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Brandstädt, A., Le, V.B., Spinrad, J.P.: Graph classes: a survey. SIAM, Philadelphia (1999)CrossRefMATHGoogle Scholar
  4. 4.
    Nikolopoulos, S., Ioannidou, K.: The longest path problem is polynomial on cocomparability graphs. In: Thilikos, D.M. (ed.) WG 2010. LNCS, vol. 6410, pp. 27–38. Springer, Heidelberg (2010)Google Scholar
  5. 5.
    Golumbic, M.C.: Algorithmic Graph Theory and Perfect Graphs. Annals of Discrete Mathematics, vol. 57. North-Holland Publishing Co., Amsterdam (2004)MATHGoogle Scholar
  6. 6.
    Ioannidou, K., Mertzios, G.B., Nikolopoulos, S.D.: The longest path problem is polynomial on interval graphs. In: Královič, R., Niwiński, D. (eds.) MFCS 2009. LNCS, vol. 5734, pp. 403–414. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  7. 7.
    Spinrad, J., Brandstädt, A., Stewart, L.: Bipartite permutation graphs. Discrete Appl. Math. 18(3), 279–292 (1987)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Uehara, R., Uno, Y.: Efficient algorithms for the longest path problem. In: Fleischer, R., Trippen, G. (eds.) ISAAC 2004. LNCS, vol. 3341, pp. 871–883. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  9. 9.
    Uehara, R., Uno, Y.: On computing longest paths in small graph classes. Int. J. Found. Comput. Sci. 18(5), 911–930 (2007)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Uehara, R., Valiente, G.: Linear structure of bipartite permutation graphs and the longest path problem. Inf. Process. Lett. 103(2), 71–77 (2007)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Esha Ghosh
    • 1
  • N. S. Narayanaswamy
    • 1
  • C. Pandu Rangan
    • 1
  1. 1.Dept. of Computer Science and EngineeringIIT MadrasChennaiIndia

Personalised recommendations