4-Connected Triangulations on Few Lines

We show that 4-connected plane triangulations can be redrawn such that edges are represented by straight segments and the vertices are covered by a set of at most $\sqrt{2n}$ lines each of them horizontal or vertical. The same holds for all subgraphs of such triangulations. The proof is based on a corresponding result for diagrams of planar lattices which makes use of orthogonal chain and antichain families.