Forcing the Wheel

Abstract

Over the past 10 years, there has been considerable interest in exploring questions connecting dimension for posets with graph theoretic properties of their cover graphs and order diagrams, especially with the concepts of planarity and treewidth. Joret and Micek conjectured that if $P$ is a poset with a planar cover graph, then the dimension of $P$ is bounded in terms of the number of minimal elements of $P$ and the treewidth of the cover graph of $P$. We settle this conjecture in the affirmative by strengthening a recent breakthrough result [14] by Blake, Micek, and Trotter, who proved that for each poset $P$ admitting a planar cover graph and a unique minimal element we have $\mathrm{dim}(P) \leq 2 \mathrm{se}(P) + 2$, namely, we prove that $\mathrm{dim}(P) \leq 2 \mathrm{wheel}(P) + 2$.