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Abstract

For a graph $G$, a vertex subset $S$ is called a maximum generalized $k$-independent set if the induced subgraph $G[S]$ does not contain a $k$-tree as its subgraph, and the subset has maximum cardinality. The generalized $k$-independence number of $G$, denoted as $\alpha_k(G)$, is the number of vertices in a maximum generalized $k$-independent set of $G$. For a graph $G$ with $n$ vertices, $m$ edges, $c$ connected components, and $c_1$ induced cycles of length 1 modulo 3, Bock et al. [J. Graph Theory 103 (2023) 661-673] showed that $\alpha_3(G)\geq n-\frac{1}{3}(m+c+c_1)$ and identified the extremal graphs in which every two cycles are vertex-disjoint. Li and Zhou [Appl. Math. Comput. 484 (2025) 129018] proved that if $G$ is a tree with $n$ vertices, then $\alpha_4(G) \geq \frac{3}{4}n$. They also presented all the corresponding extremal trees. In this paper, for a general graph $G$ with $n$ vertices, it is proved that $\alpha_4(G)\geq \frac{3}{4}(n-\omega(G))$ by using a different approach, where $\omega(G)$ denotes the dimension of the cycle space of $G$. The graphs whose generalized $4$-independence number attains the lower bound are characterized completely. This represents a logical continuation of the work by Bock et al. and serves as a natural extension of the result by Li and Zhou.