\[l-1\] Let $G=(V,E)$ be a near-bipartite graph of diameter $2$ that contains a vertex $u$ such that $G-u$ is bipartite. Then it is possible to find a minimum independent feedback vertex set of $G$ in polynomial time.

We can partition $V\setminus\{u\}$ into four independent sets $S_1$, $S_2$, $T_1$, $T_2$ (some of which might be empty) such that

(i) \[prop:i\]$S_1\cup S_2$ and $T_1\cup T_2$ form bipartition classes of $G-u$;

(ii) \[prop:ii\]$u$ is adjacent to every vertex of $S_1\cup T_1$; and

(iii) \[prop:iii\]$u$ is non-adjacent to every vertex of $S_2\cup T_2$.

Moreover, as $G$ has diameter $2$, it follows that given a vertex of $S_2$ (respectively, $T_2$) and a vertex of $T_1\cup T_2$ (respectively, $S_1\cup S_2$), these two vertices must either be adjacent or have a common neighbour. As the latter is not possible, we deduce that

(i) \[prop:iv\]every vertex of $S_2$ is adjacent to every vertex of $T_1\cup T_2$, and every vertex of $T_2$ is adjacent to every vertex of $S_1\cup S_2$ (see also  \[fig:STu\]).

=\[circle,draw=black, fill=black, minimum size=5pt, inner sep=1pt\] =\[draw,black\]

///in [ (0,-2)/T/1/below, (0,2)/S/1/above, (5,-2)/T/2/below, (5,2)/S/2/above]{}

(0.6,-0.4) rectangle (3.4,0.4); (1) at (1,0) ; (2) at (2,0) ; (3) at (3,0) ; at (2,0) [$\name_\number$]{};

\(u) at (0,0) ;

/in 