Skip to main content
SpringerLink
Log in

Canonical Antichains of Unit Interval and Bipartite Permutation Graphs

  • Published:
Order Aims and scope Submit manuscript

Abstract

We study unit interval graphs and bipartite permutation graphs partially ordered by the induced subgraph relation. It is known that neither of these classes is well-quasi-ordered, since each of them contains an infinite antichain. We show that in both cases the antichains are canonical in the sense that any subclass of unit interval or bipartite permutation graphs containing only finitely many graphs from the respective antichain is well-quasi-ordered.

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

  1. Asdre, K., Nikolopoulos, S.D.: NP-completeness results for some problems on subclasses of bipartite and chordal graphs. Theor. Comput. Sci. 381, 248–259 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  2. Benzaken, C., Hammer, P.L., de Werra, D.: Split graphs of Dilworth number 2. Discrete Math. 55, 123–127 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bogart, K.P., West, D.B.: A short proof that proper = unit. Discrete Math. 201, 21–23 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bodlaender, H.L., Kloks, T., Niedermeier, R.: Simple MAX-CUT for unit interval graphs and graphs with few P 4s. Electron. Notes Discrete Math. 3, 1–8 (1999)

    Article  MathSciNet  Google Scholar 

  5. Brandstädt, A., Lozin, V.V.: On the linear structure and clique-width of bipartite permutation graphs. Ars Comb. 67, 273–289 (2003)

    MATH  Google Scholar 

  6. Chang, R.-Sh.: Jump number maximization for proper interval graphs and series-parallel graphs. Inf. Sci. 115, 103–122 (1999)

    Article  MATH  Google Scholar 

  7. Chen, Ch., Chang, C.-C., Chang, G.J.: Proper interval graphs and the guard problem. Discrete Math. 170, 223–230 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  8. Damaschke, P.: Induced subgraphs and well-quasi-ordering. J. Graph Theory 14, 427–435 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  9. Ding, G.: Subgraphs and well-quasi-ordering. J. Graph Theory 16, 489–502 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  10. Ding, G.: Stable sets versus independent sets. Discrete Math. 117, 73–87 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  11. Ding, G.: Chordal graphs, interval graphs, and wqo. J. Graph Theory 28, 105–114 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  12. Ding, G.: On canonical antichains. Discrete Math. 309, 1123–1134 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  13. de Figueiredo, C.M.H., Meidanis, J., de Mello, C.P., Ortiz, C.: Decompositions for the edge colouring of reduced indifference graphs. Theor. Comput. Sci. 297, 145–155 (2003)

    Article  MATH  Google Scholar 

  14. Hell, P., Shamir, R., Sharan, R.: A fully dynamic algorithm for recognizing and representing proper interval graphs. SIAM J. Comput. 31, 289–305 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  15. Higman, G.: Ordering by divisibility of abstract algebras. Proc. Lond. Math. Soc. 2, 326–336 (1952)

    Article  MathSciNet  MATH  Google Scholar 

  16. Kaplan, H., Shamir, R.: Pathwidth, bandwidth, and completion problems to proper interval graphs with small cliques. SIAM J. Comput. 25, 540–561 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  17. Lozin, V.V., Rudolf, G.: Minimal universal bipartite graphs. Ars Comb. 84, 345–356 (2007)

    MathSciNet  MATH  Google Scholar 

  18. Lozin, V.V.: Minimal classes of graphs of unbounded clique-width. Ann. Comb. (accepted)

  19. Marx, D.: Precoloring extension on unit interval graphs. Discrete Appl. Math. 154, 995–1002 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  20. Petkovšek, M.: Letter graphs and well-quasi-order by induced subgraphs. Discrete Math. 244, 375–388 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  21. Roberts, F.: Indifference graphs. Proof techniques in graph theory. In: Proc. Second Ann Arbor Graph Theory Conf., Ann Arbor, Mich., 1968, pp. 139–146 (1969)

  22. Robertson, N., Seymour, P.D.: Graph Minors. XX. Wagner’s conjecture. J. Comb. Theory Ser. B 92, 325–357 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  23. Spinrad, J., Brandstädt, A., Stewart, L.: Bipartite permutation graphs. Discrete Appl. Math. 18(3), 279–292 (1987). doi:10.1016/S0166-218X(87)80003-3

    Article  MathSciNet  MATH  Google Scholar 

  24. Srikant, R., Sundaram, R., Sher Singh, K., Pandu Rangan, C.: Optimal path cover problem on cluster graphs and bipartite permutation graphs. Theor. Comput. Sci. 115, 351–357 (1993)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. DIMAP and Mathematics Institute, University of Warwick, Coventry, CV4 7AL, UK

    Vadim V. Lozin & Colin Mayhill

Authors
  1. Vadim V. Lozin
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Colin Mayhill
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Vadim V. Lozin.

Additional information

Research supported by the Centre for Discrete Mathematics and Its Applications (DIMAP), University of Warwick.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Lozin, V.V., Mayhill, C. Canonical Antichains of Unit Interval and Bipartite Permutation Graphs . Order 28, 513–522 (2011). https://doi.org/10.1007/s11083-010-9188-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11083-010-9188-7

Keywords

Search

Navigation