  Every triangle-free planar graph on n vertices has an independent set of size
at least (n+1)/3, and this lower bound is tight. We give an algorithm that,
given a triangle-free planar graph G on n vertices and an integer k>=0, decides
whether G has an independent set of size at least (n+k)/3, in time
2^{O(sqrt{k})}n. Thus, the problem is fixed-parameter tractable when
parameterized by k. Furthermore, as a corollary of the result used to prove the
correctness of the algorithm, we show that there exists epsilon>0 such that
every planar graph of girth at least five on n vertices has an independent set
of size at least n/(3-epsilon).
