A sharp lower bound on the generalized 4-independence number

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 αₖ(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 c₁ induced cycles of length 1 modulo 3, Bock et al. [J. Graph Theory 103 (2023) 661-673] showed that α₃(G)≥ n-1/3(m+c+c₁) 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 α₄(G) ≥ 3/4n. They also presented all the corresponding extremal trees. In this paper, for a general graph G with n vertices, it is proved that α₄(G)≥ 3/4(n-ω(G)) by using a different approach, where $ω(G)$ 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.