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 GS 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-13 (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) 34n. 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) 34 (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.