In this paper we study the area requirements of planar greedy drawings of
triconnected planar graphs. Cao, Strelzoff, and Sun exhibited a family H
of subdivisions of triconnected plane graphs and claimed that every planar
greedy drawing of the graphs in H respecting the prescribed plane
embedding requires exponential area. However, we show that every n-vertex
graph in H actually has a planar greedy drawing respecting the
prescribed plane embedding on an O(n)×O(n) grid. This reopens the
question whether triconnected planar graphs admit planar greedy drawings on a
polynomial-size grid. Further, we provide evidence for a positive answer to the
above question by proving that every n-vertex Halin graph admits a planar
greedy drawing on an O(n)×O(n) grid. Both such results are obtained by
actually constructing drawings that are convex and angle-monotone. Finally, we
consider α-Schnyder drawings, which are angle-monotone and hence greedy
if α≤30∘, and show that there exist planar triangulations for
which every α-Schnyder drawing with a fixed α<60∘ requires
exponential area for any resolution rule