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A feedback vertex set of a graph is a subset of vertices intersecting allcycles. We provide tight upper bounds on the size of a minimum feedback vertexset in planar graphs of girth at least five. We prove that if $G$ is aconnected planar graph of girth at least five on $n$ vertices and $m$ edges,then $G$ has a feedback vertex set of size at most $\frac{2m-n+2}{7}$. ByEuler's formula, this implies that $G$ has a feedback vertex set of size atmost $\frac{m}{5}$ and $\frac{n-2}{3}$. These results not only improve a resultof Dross, Montassier and Pinlou and confirm the girth-5 case of one of theirconjectures, but also make the best known progress towards a conjecture ofKowalik, Lu\v{z}ar and \v{S}krekovski and solves the subcubic case of theirconjecture. An important step of our proof is providing an upper bound on thesize of minimum feedback vertex sets of subcubic graphs with girth at leastfive with no induced subdivision of members of a finite family of non-planargraphs.

Minimum Size of Feedback Vertex Sets of Planar Graphs of Girth at least Five