Size of the Largest Induced Forest in Subcubic Graphs of Girth at least Four and Five

In this paper, we address the maximum number of vertices of induced forests in subcubic graphs with girth at least four or five. We provide a unified approach to prove that every 2-connected subcubic graph on $n$ vertices and $m$ edges with girth at least four or five, respectively, has an induced forest on at least $n-\frac{2}{9}m$ or $n-\frac{1}{5}m$ vertices, respectively, except for finitely many exceptional graphs. Our results improve a result of Liu and Zhao and are tight in the sense that the bounds are attained by infinitely many 2-connected graphs. Equivalently, we prove that such graphs admit feedback vertex sets with size at most $\frac{2}{9}m$ or $\frac{1}{5}m$, respectively. Those exceptional graphs will be explicitly constructed, and our result can be easily modified to drop the 2-connectivity requirement

On cycle packings and feedback vertex sets

For a graph $G$, let $\mathbf{fvs}$ and $\mathbf{cp}$ denote the minimum size of a feedback vertex set in $G$ and the maximum size of a cycle packing in $G$, respectively. Kloks, Lee, and Liu conjectured that $\mathbf{fvs}(G)\le 2\,\mathbf{cp}(G)$ if $G$ is planar. They proved a weaker inequality, replacing $2$ by $5$. We improve this, replacing $5$ by $3$, and verifying the resulting inequality for graphs embedded in surfaces of nonnegative Euler characteristic. We also generalize to arbitrary surfaces. We show that, if a graph $G$ embeds in a surface of Euler characteristic $c\le 0$, then $\mathbf{fvs}(G)\le 3\,\mathbf{cp}(G) + 103(-c)$. Lastly, we consider what the best possible bound on $\mathbf{fvs}$ might be, and give some open problems