The Dimension of Posets with Planar Cover Graphs

摘要

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 is a poset with a planar comparability graph, then the dimension of is at most four. We also show that if has an outerplanar cover graph, then the dimension of is at most four. Finally, if has an outerplanar cover graph and the height of is two, then the dimension of is at most three. These three inequalities are all best possible.