graphs and $P_4$-free graphs (the latter result generalising the result of [@BBKNP13] for [Near-Bipartiteness]{} on $P_4$-free graphs). In [@BDFJP17b] we proved that finding a minimum independent feedback vertex set is polynomial-time solvable even for $P_5$-free graphs. We refer to [@AGSS16; @MPRS12] for FPT algorithms with parameter $k$ for finding an independent feedback vertex set of size at most $k$.

The [*distance*]{} between two vertices $u$ and $v$ in a graph $G$ is the length (number of edges) of a shortest path between $u$ and $v$. The [*diameter*]{} of a graph $G$ is the maximum distance between any two vertices in $G$. In addition to their results for graphs of bounded maximum degree, Yang and Yuan [@YY06] proved that [Near-Bipartiteness]{} is polynomial-time solvable for graphs of diameter at most $2$ and [[NP]{}]{}-complete for graphs of diameter at most $4$. They asked the following question, which was also posed by Brandstädt et al. [@BBKNP13]:

[*What is the complexity of [Near-Bipartiteness]{} for graphs of diameter $3$?*]{}

[**Our Results.**]{} We complete the complexity classifications of [Near-Bipartiteness]{} and [Independent Feedback Vertex Set]{} for graphs of bounded diameter. In particular, we prove that [Near-Bipartiteness]{} is [[NP]{}]{}-complete for graphs of diameter $3$, which answers the above question. 