Skip to main content
SpringerLink
Log in

A characterization of chain probe graphs

  • Published:
Annals of Operations Research Aims and scope Submit manuscript

Abstract

A chain probe graph is a graph that admits an independent set S of vertices and a set F of pairs of elements of S such that G+F is a chain graph (i.e., a 2K 2-free bipartite graph). We show that chain probe graphs are exactly the bipartite graphs that do not contain as an induced subgraph a member of a family of six forbidden subgraphs, and deduce an O(n 2) recognition algorithm.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Berry, A., Golumbic, M. C., & Lipshteyn, M. (2004). Two tricks to triangulate chordal probe graphs in polynomial time. In Proc. SODA’04, 962–969.

  • Berry, A., Golumbic, M. C., & Lipshteyn, M. (2007). Cycle-bicolorable graphs and triangulating chordal probe graphs. SIAM Journal on Discrete Mathematics, 21, 573–591.

    Article  Google Scholar 

  • Brandstädt, A., Le, V., & Spinrad, J. P. (1999) Graph classes: A survey. SIAM Monographs on Discrete Mathematics and Applications, Philadelphia.

  • Chandler, D. D., Chang, M.-S., Kloks, T., Le, V. B., & Peng, S.-L. (2008). Probe Ptolemaic graphs. In Proc. COCOON 2008. Lecture notes in computer science (Vol. 5092, pp. 468–477).

  • Chang, M.-S., Kloks, T., Kratsch, D., Liu, J., & Peng, S.-L. (2005). On the recognition of probe graphs of some self-complementary classes of perfect graphs. In Proc. COCOON 2005. Lecture notes in computer science (Vol. 3595, pp. 808–817).

  • Chvátal, V., & Hammer, P. (1977). Aggregation of inequalities integer programming. Annals of Discrete Mathematics, 1, 145–162.

    Article  Google Scholar 

  • Golumbic, M. C. (2004). Algorithmic graph theory and perfect graphs (2nd edn.) Annals of discrete math., Vol. 57. Amsterdam: Elsevier.

    Google Scholar 

  • Golumbic, M. C., & Lipshteyn, M. (2003). Chordal probe graphs (extended abstract). In Proceedings of WG’03. Lecture notes in computer science (Vol. 2880, pp. 249–260).

  • Golumbic, M. C., & Lipshteyn, M. (2004). Chordal probe graphs. Discrete Applied Mathematics, 143, 221–237.

    Article  Google Scholar 

  • Golumbic, M. C., & Trenk, A. N. (2004). Tolerance graphs. Cambridge: Cambridge University Press.

    Book  Google Scholar 

  • Golumbic, M. C., Kaplan, H., & Shamir, R. (1995). Graph sandwich problems. Journal of Algorithms, 19, 449–473.

    Article  Google Scholar 

  • Hammer, P. L., Peled, U. N., & Sun, X. (1990). Difference graphs. Discrete Applied Mathematics, 28, 35–44.

    Article  Google Scholar 

  • Johnson, J. L., & Spinrad, J. P. (2001). A polynomial time recognition algorithm for probe interval graphs. In Proc. 12th annual ACM-SIAM symp. on disc. algorithms (SODA) (pp. 477–486).

  • Le, V. B., & de Ridder, E. (2007a). Probe split graphs. Discrete Mathematics and Theoretical Computer Science, 9, 207–238.

    Google Scholar 

  • Le, V. B., & de Ridder, E. (2007b). Characterisations and linear-time recognition of probe cographs. In Proc. WG 2007. Lecture notes in computer science (Vol. 4769, pp. 226–237).

  • Mahadev, N. V. R., & Peled, U. N. (1995). Threshold graphs and related topics. Annals of discrete mathematics (Vol. 56). Amsterdam: Elsevier.

    Google Scholar 

  • McConnell, R. M., & Spinrad, J. P. (2002). Construction of probe interval models. In Proc. 13th annual ACM-SIAM Symp. on disc. algorithms (SODA) (pp. 866–875).

  • McMorris, F. R., Wang, C., & Zhang, P. (1998). On probe interval graphs. Discrete Applied Mathematics, 88, 315–324.

    Article  Google Scholar 

  • Przulj, N., & Corneil, D. G. (2005). 2-Tree probe interval graphs have a large obstruction set. Discrete Applied Mathematics, 150, 216–231.

    Article  Google Scholar 

  • Sheng, L. (1999). Cycle free probe interval graphs. Congressus Numerantium, 140, 33–42.

    Google Scholar 

  • Yannakakis, M. (1981). Node-deletion problems on bipartite graphs. SIAM Journal on Computing, 10, 310–327.

    Article  Google Scholar 

  • Zhang, P. (1994). Probe interval graphs and their application to physical mapping of DNA. Manuscript.

  • Zhang, P., Schon, E. A., Fischer, S. G. Cayanis, E. Weiss, J. Kistler, S. & Bourne, P.E. (1999a). An algorithm based on graph theory for the assembly of contigs in physical mapping of DNA. CABIOS, 10, 309–317.

    Google Scholar 

  • Zhang, P., Ye, X., Liao, L., Russo, J., & Fischer, S. G. (1999b). Integrated mapping package—a physical mapping software tool kit. Genomics, 55, 78–87.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Caesarea Rothschild Institute and Department of Computer Science, University of Haifa, Haifa, Israel

    Martin C. Golumbic

  2. C.N.R.S., Laboratoire G-SCOP, Grenoble, France

    Frédéric Maffray

  3. Laboratoire G-SCOP, Grenoble, France

    Grégory Morel

Authors
  1. Martin C. Golumbic
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Frédéric Maffray
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Grégory Morel
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Frédéric Maffray.

Additional information

This research was made possible by the joint French–Israeli project Recognition, Decomposition, and Optimization Problems in Graph Theory.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Golumbic, M.C., Maffray, F. & Morel, G. A characterization of chain probe graphs. Ann Oper Res 188, 175–183 (2011). https://doi.org/10.1007/s10479-009-0584-6

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10479-009-0584-6

Keywords

Search

Navigation