The alpha forest-cut conjecture

Let $G$ be a connected graph on $n$ vertices, and let $\alpha$ satisfy

$$2\leq \alpha\leq 3.$$

A forest cut is a vertex cut whose induced subgraph is a forest.

$\alpha$-FC conjecture. If $G$ has no forest cut, then

$$e(G)\geq \alpha(n-3)+3.$$

This parameterized conjecture unifies lower bounds for connected graphs without forest cuts and is introduced to organize the structural arguments in the paper. The paper proves the case corresponding to its bound $\frac94n-\frac{15}{4}$, but does not establish the full parameterized assertion.