  It is proven that every set $S$ of distinct points in the plane with
cardinality $\lceil \frac{\sqrt{\log_2 n}-1}{4} \rceil$ can be a subset of the
vertices of a crossing-free straight-line drawing of any planar graph with $n$
vertices. It is also proven that if $S$ is restricted to be a one-sided convex
point set, its cardinality increases to $\lceil \sqrt[3]{n} \rceil$. The proofs
are constructive and give rise to O(n)-time drawing algorithms. As a part of
our proofs, we show that every maximal planar graph contains a large induced
biconnected outerplanar graphs and a large induced outerpath (an outerplanar
graph whose weak dual is a path).
