, Volume 33, Issue 2, pp 333–346 | Cite as

Planar Posets, Dimension, Breadth and the Number of Minimal Elements

  • William T. Trotter
  • Ruidong Wang


In recent years, researchers have shown renewed interest in combinatorial properties of posets determined by geometric properties of its order diagram and topological properties of its cover graph. In most cases, the roots for the problems being studied today can be traced back to the 1970’s, and sometimes even earlier. In this paper, we study the problem of bounding the dimension of a planar poset in terms of the number of minimal elements, where the starting point is the 1977 theorem of Trotter and Moore asserting that the dimension of a planar poset with a single minimal element is at most 3. By carefully analyzing and then refining the details of this argument, we are able to show that the dimension of a planar poset with t minimal elements is at most 2t + 1. This bound is tight for t = 1 and t = 2. But for t ≥ 3, we are only able to show that there exist planar posets with t minimal elements having dimension t + 3. Our lower bound construction can be modified in ways that have immediate connections to the following challenging conjecture: For every d ≥ 2, there is an integer f(d) so that if P is a planar poset with dim(P) ≥ f(d), then P contains a standard example of dimension d. To date, the best known examples only showed that the function f, if it exists, satisfies f(d) ≥ d + 2. Here, we show that lim d→∞ f(d)/d ≥ 2.


Planar poset Dimension 


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  1. 1.
    Baker, K.: Dimension, join-independence and breadth in partially ordered sets, unpublished manuscriptGoogle Scholar
  2. 2.
    Baker, K., Fishburn, P.C., Roberts, F.R.: Partial orders of dimension 2, interval orders and interval graphs. Networks 2, 11–28 (1971)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Di Battista, G., Liu, W.-P., Rival, I.: Bipartite graphs, upward drawings, and planarity, Inform. Process. Lett. 36, 317–322 (1990)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Biró, C., Keller, M.T., Young, S.J.: Posets with cover graph of pathwidth two have bounded dimension, Order, to appear but published online as of July 3, 2015 at doi: 10.1007/s11083-015-9359-7
  5. 5.
    Brightwell, G.R.: On the complexity of diagram testing. Order 10, 297–303 (1993)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Felsner, S., Trotter, W.T.: Dimension, graph and hypergraph coloring. Order 17, 167–177 (2000)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Felsner, S., Li, C.M., Trotter, W.T.: Adjacency posets of planar graphs. Discret. Math. 310, 1097–1104 (2010)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Felsner, S., Trotter, W.T., Wiechert, V.: The dimension of posets with planar cover graphs, Graphs Combin. in press and available on-line at Springer Japan doi: 10.1007/s0073-014-1430-4.
  9. 9.
    Fishburn, P.C.: Intransitive difference with unequal indifference intervals. J. Math. Psychol. 7, 144–149 (1970)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Füredi, Z., Hajnal, P., Rödl, V., Trotter, W.T.: Interval orders and shift graphs, in Sets, Graphs and Numbers. Colloq. Math. Soc. Janos Bolyai 60, 297–313 (1991)MATHGoogle Scholar
  11. 11.
    Garg, A., Tamassia, R.: On the computational complexity of upward and rectilinear planarity testing, SIAM. J. Comput. 31, 601–625 (2001)MathSciNetMATHGoogle Scholar
  12. 12.
    Hopcroft, J., Tarjan, R.E.: Efficient planarity testing. J. Assn. for Comp. Mach. 21, 549–568 (1974)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Joret, G., Micek, P., Milans, K., Trotter, W.T., Walczak, B., Wang, R.: Tree-width and dimension, Combinatorica, to appear but published online as of February 15, 2015 at doi: 10.1007/s00493-014-3081-8.
  14. 14.
    Joret, G., Micek, P., Trotter, W.T., Wang R., Wiechert, V.: On the Dimension of Posets with Cover Graphs of Tree-width 2, submitted, available on the ArXiv:1406.3397v2
  15. 15.
    Kelly, D.: The 3-irreducible partially ordered sets, Canad. J. Math. 29, 367–383 (1977)MathSciNetMATHGoogle Scholar
  16. 16.
    Kelly, D.: On the dimension of partially ordered sets. Discret. Math. 35, 135–156 (1981)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Kierstead, H., Trotter, W.T.: The number of depth-first searches of an ordered set. Order 6, 295–303 (1989)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Kierstead, H., Trotter, W.T., et al.: Super-greedy linear extensions of ordered sets, in Combinatorial Mathematics. Ann. N. Y. Acad. Sci. 555, 262–271 (1989)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Kierstead, H., Trotter, W.T.: Interval orders and dimension. Discret. Math. 213, 179–188 (2000)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Kierstead, H., Trotter, W.T., Zhou, B.: Representing an ordered set as the intersection of super-greedy linear extensions. Order 4, 293–311 (1987)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Micek, P., Wiechert, V.: Topological minors of cover graphs and dimension, submittedGoogle Scholar
  22. 22.
    Nešetřil, J., Rödl, V.: Complexity of diagrams. Order 3, 321–330 (1987). 321?-330. Corrigendum: Order 10 (1993), 393MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Streib, N., Trotter, W.T.: Dimension and height for posets with planar cover graphs, European. J. Comb. 35, 474–489 (2014)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Trotter, W.T.: Dimension of the crown \({S_{n}^{k}}\). Discret. Math. 8, 85–103 (1974)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Trotter, W.T.: Order preserving embeddings of aographs. Congressus Numerantium 642, 572–579 (1978)MathSciNetMATHGoogle Scholar
  26. 26.
    Trotter, W.T.: Combinatorial problems in dimension theory for partially ordered Sets, in Problemes Combinatoires et Theorie des Graphes. Colloq. Int. C. N. R. S. 260, 403–406 (1978)MathSciNetGoogle Scholar
  27. 27.
    Trotter, W.T.: Combinatorics and Partially Ordered Sets: Dimension Theory. The Johns Hopkins University Press, Baltimore (1992)MATHGoogle Scholar
  28. 28.
    Trotter, W.T.: Partially ordered sets, in Handbook of Combinatorics. In: Graham, R.L., Grötschel, M., Lovász, L. (eds.) , pp 433–480. Elsevier, Amsterdam (1995)Google Scholar
  29. 29.
    Trotter, W.T., Moore, J.I.: Some theorems on graphs and posets. Discret. Math. 15, 79–84 (1976)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Trotter, W.T., Moore, J.I.: The dimension of planar posets. J. Combin. Theory Ser. B 21, 51–67 (1977)MathSciNetMATHGoogle Scholar
  31. 31.
    Walczak, B.: Minors and dimension, Proc. 26th Annual ACM-SIAM Symposium on Discrete Algorithms, 2015, 1698-1707Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  1. 1.School of MathematicsGeorgia Institute of TechnologyAtlantaUSA

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