Abstract
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(root 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).
We further give an algorithm that, given a planar graph G of maximum degree 4 on n vertices and an integer k >= 0, decides whether G has an independent set of size at least (n + k)/4, in time 2(O(root k)) n.