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 nβ92βm or nβ51βm 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 92βm or 51βm, respectively.
Those exceptional graphs will be explicitly constructed, and our result can be
easily modified to drop the 2-connectivity requirement