  Consider the following problem: Given a planar graph $G$, what is the maximum
number $p$ such that $G$ has a planar straight-line drawing with $p$ collinear
vertices? This problem resides at the core of several graph drawing problems,
including universal point subsets, untangling, and column planarity. The
following results are known for it: Every $n$-vertex planar graph has a planar
straight-line drawing with $\Omega(\sqrt{n})$ collinear vertices; for every
$n$, there is an $n$-vertex planar graph whose every planar straight-line
drawing has $O(n^\sigma)$ collinear vertices, where $\sigma<0.986$; every
$n$-vertex planar graph of treewidth at most two has a planar straight-line
drawing with $\Theta(n)$ collinear vertices. We extend the linear bound to
planar graphs of treewidth at most three and to triconnected cubic planar
graphs. This (partially) answers two open problems posed by Ravsky and
Verbitsky [WG 2011:295--306]. Similar results are not possible for all bounded
treewidth planar graphs or for all bounded degree planar graphs. For planar
graphs of treewidth at most three, our results also imply asymptotically tight
bounds for all of the other above mentioned graph drawing problems.
