Planar Posets, Dimension, Breadth and the Number of Minimal Elements
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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.
KeywordsPlanar poset Dimension
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- 1.Baker, K.: Dimension, join-independence and breadth in partially ordered sets, unpublished manuscriptGoogle Scholar
- 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
- 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.
- 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.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
- 21.Micek, P., Wiechert, V.: Topological minors of cover graphs and dimension, submittedGoogle Scholar
- 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
- 31.Walczak, B.: Minors and dimension, Proc. 26th Annual ACM-SIAM Symposium on Discrete Algorithms, 2015, 1698-1707Google Scholar