vertices of $I$ are adjacent. For $u\in V$, we let $G-u$ denote the graph obtained from $G$ after deleting the vertex $u$ (and its incident edges). A graph is [*complete bipartite*]{} if its vertex set can be partitioned into two independent sets $S$ and $T$ such that there is an edge between every vertex of $S$ and every vertex of $T$. If $S$ or $T$ has size $1$, the graph is also called a [*star*]{}.

\[t-yy06\] A graph $G=(V,E)$ of diameter $2$ is near-bipartite if and only if one of the following two conditions holds:

(i) \[cond:i\]there exists a vertex $u$ such that $G-u$ is bipartite; or

(ii) \[cond:ii\]there exists a set $X$, $4\leq |X|\leq 5$, such that $(A_X,V\setminus A_X)$ is a near-bipartite decomposition.

As noted in [@YY06], Theorem \[t-yy06\] can be used to solve [Near-Bipartiteness]{} in polynomial time for graphs of diameter $2$, as conditions \[cond:i\] and \[cond:ii\] can be checked in polynomial time. However, Theorem \[t-yy06\] does not tell us how to determine the size of a minimum independent feedback vertex set.

In order to find a minimum independent feedback vertex set, we will distinguish between the two cases of Theorem \[t-yy06\]. This leads to two corresponding lemmas.

