Sharp lower bounds and extremal graphs for the generalized $k$-independence number

A vertex set $S$ is a generalized $k$-independent set if the induced subgraph $G[S]$ contains no tree on $k$ vertices. The generalized $k$-independence number $\alpha_k(G)$ is the maximum size of such a set. For a tree $T$ with $n$ vertices, Bock et al. [J. Graph Theory 103 (2023) 661-673] and Li et al. [Taiwanese J. Math. 27 (2023) 647-683] independently showed that $\alpha_3(G)\geq \frac{2}{3}n$ and identified the extremal trees that attain this lower bound. Subsequently, Li and Zhou [Appl. Math. Comput. 484 (2025) 129018] established that $\alpha_4(T) \geq \frac{3}{4}n$ and they further characterized all trees achieving this bound. This result was recently extended by Huang, who proved that $\alpha_4(G)\geq \frac{3}{4}(n-\omega(G))$ holds for every $n$-vertex graph, where $\omega(G)$ denotes the dimension of the cycle space of $G.$ The extremal graphs attaining this lower bound were also fully characterized. Based on these findings, Huang proposed a conjecture concerning a lower bound for $\alpha_k(G)\ (k\geq2)$ together with the corresponding extremal graphs, which naturally generalizes all the aforementioned results. In this paper, we confirm this conjecture here. We further quantify strict improvements over this bound when the equality conditions fail, and we provide a linear-time algorithm that constructs a generalized $k$-independent set of size at least $\left\lceil\frac{k-1}{k}\left(n-\omega(G)\right)\right\rceil$.