The dimension of a poset is one if and only if it is a linear order, i.e., there are no incomparable pairs. When P is not a linear order, the dimension of P is ...

Jul 2, 2013 · Trotter conjectured in [9] that a poset with a planar cover graph has dimension which can be bounded in terms of its height. The primary ...

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 ...

Jun 30, 2019 · We show that height~h posets that have planar cover graphs have dimension O(h^6). Previously, this upper bound was 2^{O(h^3)}.

We show that the dimension of a planar poset is bounded as a function of its height. In fact this statement holds for all posets with planar cover graphs.

Jun 18, 2012 · Let D be a non-crossing drawing of a planar multigraph G, and let P be the vertex-edge-face incidence poset determined by D. Then dim(P) ≤ 4.

Abstract. We show that for each positive integer h, there exists a least positive integer c(h) so that if P is a poset having a planar cover graph and ...

For a positive integer h, let d(h) be the largest integer t for which there exists a poset P of height h and dimension t such that the cover graph of P is ...

It is proved that every planar poset of height $P$ of height h has dimension at most $192h+96$ and this improves on previous exponential bounds and is best ...

Oct 22, 2024 · The class of planar graphs does have bounded expansion, and their proof implied that the dimension of a height-h poset with a planar cover graph ...