Chernyshev–Rauch–Rautenbach conjecture on sparse graphs and forest cuts

Let $G$ be a graph of order $n$, and let a forest cut mean a vertex cut $M$ such that the subgraph $G|_M$ induced by the cut vertices is a forest.

Chernyshev–Rauch–Rautenbach conjecture. If $G$ has fewer than $3n-6$ edges, then $G$ has a forest cut.

The conjecture is known for planar graphs, while the paper improves previously proved linear edge bounds to fewer than $(19n-28)/8$ edges; the sharp threshold $3n-6$ remains unresolved in general.