Bipartite graphs, upward drawings, and planarity
Information Processing Letters
The order dimension of convex polytopes
SIAM Journal on Discrete Mathematics
Handbook of combinatorics (vol. 1)
Journal of the ACM (JACM)
On the Computational Complexity of Upward and Rectilinear Planarity Testing
SIAM Journal on Computing
The Order Dimension of Planar Maps
SIAM Journal on Discrete Mathematics
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We show that for each integer h=2, there exists a least positive integer c"h so that if P is a poset having a planar cover graph and the height of P is h, then the dimension of P is at most c"h. Trivially, c"1=2. Also, Felsner, Li and Trotter showed that c"2 exists and is 4, but their proof techniques do not seem to apply when h=3. We focus on establishing the existence of c"h, although we suspect that the upper bound provided by our proof is far from best possible. From below, a construction of Kelly is easily modified to show that c"h must be at least h+2.