, Volume 33, Issue 2, pp 195–212 | Cite as

Posets with Cover Graph of Pathwidth two have Bounded Dimension

  • Csaba Biró
  • Mitchel T. Keller
  • Stephen J. Young


Joret, Micek, Milans, Trotter, Walczak, and Wang recently asked if there exists a constant d such that if P is a poset with cover graph of P of pathwidth at most 2, then dim(P)=d. We answer this question in the affirmative by showing that d=17 is sufficient. We also show that if P is a poset containing the standard example S5 as a subposet, then the cover graph of P has treewidth at least 3.


Poset Pathwidth Cover graph Dimension 


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  1. 1.
    Barát, J., Hajnal, P., Lin, Y., Yang, A.: On the structure of graphs with path-width at most two. Studia Sci. Math. Hungar. 49(2), 211–222 (2012)MathSciNetMATHGoogle Scholar
  2. 2.
    Diestel, R.: Graph theory, 4th edn., vol. 173 of Graduate Texts in Mathematics. Springer-Verlag, New York (2010)Google Scholar
  3. 3.
    Felsner, S., Li, C.M., Trotter, W.T.: Adjacency posets of planar graphs. Discret. Math. 310(5), 1097–1104 (2010)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Felsner, S., Trotter, W.T., Wiechert, V.: The dimension of posets with planar cover graphs. To appear in Graphs Combin (2013)Google Scholar
  5. 5.
    Halin, R.: S-functions for graphs. J. Geometry 8(1–2), 171–186 (1976)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Joret, G., Micek, P., Milans, K.G., Trotter, W.T., Walczak, B., Wang, R.: Tree-width and dimension. To appear in Combinatorica (2014)Google Scholar
  7. 7.
    Joret, G., Micek, P., Trotter, W.T., Wang, R., Wiechert, V.: On the dimension of posets with cover graphs of treewidth 2. SubmittedGoogle Scholar
  8. 8.
    Kelly, D.: On the dimension of partially ordered sets. Discret. Math. 35, 135–156 (1981)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Kinnersley, N.G., Langston, M.A.: Obstruction set isolation for the gate matrix layout problem. Discrete Appl. Math. 54(2–3), 169–213 (1994). Efficient algorithms and partial k-treesMathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Robertson, N., Seymour, P.D.: Graph minors. III. Planar tree-width. J. Combin. Theory Ser. B 36(1), 49–64 (1984)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Robertson, N., Seymour, P.D.: Graph minors. XX. Wagner’s conjecture. J. Combin. Theory Ser. B 92(2), 325–357 (2004)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Spinrad, J.P.: Edge subdivision and dimension. Order 5(2), 143–147 (1988)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Streib, N., Trotter, W.T.: Dimension and height for posets with planar cover graphs. European J. Combin. 35, 474–489 (2014)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Trotter, W.T.: Combinatorics and partially ordered sets: Dimension theory. Johns Hopkins Series in the Mathematical Sciences. Johns Hopkins University Press, Baltimore (1992)Google Scholar
  15. 15.
    Trotter, W.T.: Personal communication (2013)Google Scholar
  16. 16.
    Trotter, W.T., Moore, J.I.: The dimension of planar posets. J. Combin. Theory Ser. B 22(1), 54–67 (1977)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  • Csaba Biró
    • 1
  • Mitchel T. Keller
    • 2
  • Stephen J. Young
    • 1
  1. 1.Department of MathematicsUniversity of LouisvilleLouisvilleUSA
  2. 2.Department of MathematicsWashington and Lee UniversityLexingtonUSA

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