Large Independent Sets in Triangle-Free Planar Graphs

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 $k\geq0$, 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 $\varepsilon>0$ such that every planar graph of girth at least five on $n$ vertices has an independent set of size at least $n/(3-\varepsilon)$. We further give an algorithm that, given a planar graph $G$ of maximum degree 4 on $n$ vertices and an integer $k\geq0$, decides whether $G$ has an independent set of size at least $(n+k)/4$, in time $2^{O(\sqrt{k})}n$.