Large Independent Sets in Triangle-Free Planar Graphs

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 √ 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 ε > 0 such that every planar graph of girth at least five on n vertices has an independent set of size at least n/(3− ε).