Recognition of Probe Cographs and Partitioned Probe Distance Hereditary Graphs

  • David B. Chandler
  • Maw-Shang Chang
  • Ton Kloks
  • Jiping Liu
  • Sheng-Lung Peng
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4041)


Given a class of graphs \({\cal G}\), a graph G is a probe graph of \({\cal G}\) if its vertices can be partitioned into two sets ℙ (the probes) and ℕ (nonprobes), where ℕ is an independent set, such that G can be embedded into a graph of \({\cal G}\) by adding edges between certain vertices of ℕ. If the partition of the vertices into probes and nonprobes is part of the input, then we call the graph a partitioned probe graph of \({\cal G}\). We give the first polynomial-time algorithm for recognizing partitioned probe distance-hereditary graphs. By using a novel data structure for storing a multiset of sets of numbers, the running time of this algorithm is \({O}(\mathfrak\it{n}^2)\), where \(\mathfrak\it{n}\) is the number of vertices of the input graph. We also show that the recognition of both partitioned and unpartitioned probe cographs can be done in \({O}(\mathfrak\it{n}^2)\) time.


Leaf Node Binary Tree Recognition Algorithm Interval Graph Decomposition Tree 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bandelt, H.J., Mulder, H.M.: Distance-hereditary graphs. Journal of Combinatorial Theory, Series B 41, 182–208 (1989)CrossRefMathSciNetGoogle Scholar
  2. 2.
    Berry, A., Golumbic, M.C., Lipshteyn, M.: Two tricks to triangulate chordal probe graphs in polynomial time. In: Proceedings 15th ACM–SIAM Symposium on Discrete Algorithms, pp. 962–969 (2004)Google Scholar
  3. 3.
    Brändstadt, A., Van Bang, L., Spinrad, J.P.: Graph Classes: A Survey. SIAM Monographs on Discrete Mathematics and Applications, Philadelphia (1999)Google Scholar
  4. 4.
    Chang, Jennwha, G., Kloks, T., Peng, S.-L.: Probe interval bigraphs (extended abstract). Electronic Notes in Discrete Mathematics 19, 195–201 (2005)CrossRefGoogle Scholar
  5. 5.
    Chang, Jennwha, G., Kloks, A.J.J., Liu, J., Peng, S.-L.: The PIGs full monty - a floor show of minimal separators. In: Diekert, V., Durand, B. (eds.) STACS 2005. LNCS, vol. 3404, pp. 521–532. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  6. 6.
    Chang, M.-S., Kloks, T., Kratsch, D., Liu, J., Peng, S.-L.: On the recognition of probe graphs of some self-complementary graph classes. In: Wang, L. (ed.) COCOON 2005. LNCS, vol. 3595, pp. 808–817. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  7. 7.
    Chang, Maw-Shang, Hsieh, S.Y., Chen, G.H.: Dynamic programming on distance-hereditary graphs. In: Leong, H.-V., Jain, S., Imai, H. (eds.) ISAAC 1997. LNCS, vol. 1350, pp. 344–353. Springer, Heidelberg (1997)CrossRefGoogle Scholar
  8. 8.
    Corneil, D.G., Lerchs, H., Stewart-Burlingham, L.: Complement reducible graphs. Discrete Applied Mathematics 3, 163–174 (1981)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Corneil, D.G., Perl, Y., Stewart, L.K.: A linear recognition algorithm for cographs. SIAM Journal on Computing 14, 926–934 (1985)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Damiand, G., Habib, M., Paul, C.: A simple paradigm for graph recognition: application to cographs and distance-hereditary graphs. Theoretical Computer Science 263, 99–111 (2001)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Golumbic, M.C., Kaplan, H., Shamir, R.: Graph sandwich problems. J. of Algorithms 19, 449–473 (1995)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Golumbic, M.C., Lipshteyn, M.: Chordal probe graphs. Discrete Applied Mathematics 143, 221–237 (2004)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Habib, M., Paul, C.: A simple linear time algorithm for cograph recognition. Discrete Applied Mathematics 145, 183–197 (2005)MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Howorka, E.: A characterization of distance-hereditary graphs. The Quarterly Journal of Mathematics 28, 417–420 (1977)MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Hsieh, S.-Y., Ho, C.-W., Hsu, T.-S., Ko, M.-T., Chen, G.-H.: A faster implementation of a parallel tree contraction scheme and its application on distance-hereditary graphs. Journal of Algorithms 35, 50–81 (2000)MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Johnson, J.L., Spinrad, J.: A polynomial-time recognition algorithm for probe interval graphs. In: Proceedings 12th ACM–SIAM Symposium on Discrete Algorithms, pp. 477–486 (2001)Google Scholar
  17. 17.
    McConnell, R.M., Spinrad, J.P.: Modular decomposition and transitive orientation. Discrete Mathematics 201, 189–241 (1999)MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    McConnell, R.M., Spinrad, J.: Construction of probe interval graphs. In: Proceedings 13th ACM–SIAM Symposium on Discrete Algorithms, pp. 866–875 (2002)Google Scholar
  19. 19.
    McMorris, F.R., Wang, C., Zhang, P.: On probe interval graphs. Discrete Applied Mathematics 88, 315–324 (1998)MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Mehlhorn, Kurt: Data Structures and Algorithms 1: Sorting and Searching. Monographs in Theoretical Computer Science, an EATCS series, vol. 1. Springer, Heidelberg (1984)MATHGoogle Scholar
  21. 21.
    Sheng, L.: Cycle-free probe interval graphs. Congressus Numerantium 140, 33–42 (1999)MATHMathSciNetGoogle Scholar
  22. 22.
    Zhang, P., Schon, E.A., Fisher, S.G., Cayanis, E., Weiss, J., Kistler, S., Bourne, P.E.: An algorithm based on graph theory for the assembly of contigs in physical mapping of DNA. CABIOS 10, 309–317 (1994)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • David B. Chandler
    • 1
  • Maw-Shang Chang
    • 2
  • Ton Kloks
  • Jiping Liu
    • 3
  • Sheng-Lung Peng
    • 4
  1. 1.Institute of MathematicsAcademia Sinica, NangangTaipeiTaiwan
  2. 2.Department of Computer Science and Information EngineeringNational Chung Cheng UniversityChiayiTaiwan
  3. 3.Department of Mathematics and Computer ScienceThe University of LethbridgeAlbertaCanada
  4. 4.Department of Computer Science and Information EngineeringNational Dong Hwa UniversityHualienTaiwan

Personalised recommendations