Abstract
Kelly showed that there exist planar posets of arbitrarily large dimension, and Streib and Trotter showed that the dimension of a poset with a planar cover graph is bounded in terms of its height. Here we continue the study of conditions that bound the dimension of posets with planar cover graphs. We show that if \(P\) is a poset with a planar comparability graph, then the dimension of \(P\) is at most four. We also show that if \(P\) has an outerplanar cover graph, then the dimension of \(P\) is at most four. Finally, if \(P\) has an outerplanar cover graph and the height of \(P\) is two, then the dimension of \(P\) is at most three. These three inequalities are all best possible.
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Acknowledgments
William T. Trotter would like to thank Professor Ton Kloks for raising the question of the dimension of posets with outerplanar cover graphs during the \(6{\text {th}}\) Cross Strait Conference on Graph Theory and Combinatorics. It should also be noted that the third author had independently proven Theorem 1.9, using a different approach, one that did not depend on Theorem 1.8. In particular, his argument was a direct application of Theorem 1.2. This work was done as part of his B.Sc. thesis at TU Berlin.
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Felsner, S., Trotter, W.T. & Wiechert, V. The Dimension of Posets with Planar Cover Graphs. Graphs and Combinatorics 31, 927–939 (2015). https://doi.org/10.1007/s00373-014-1430-4
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DOI: https://doi.org/10.1007/s00373-014-1430-4