  We say a null-homologous knot $K$ in a $3$--manifold $Y$ has Property G, if
the properties about the Thurston norm and fiberedness of the complement of $K$
is preserved under the zero surgery on $K$. In this paper, we will show that,
if the smooth $4$--genus of $K\times\{0\}$ (in a certain homology class) in
$(Y\times[0,1])\#N\overline{\mathbb CP^2}$, where $Y$ is a rational homology
sphere, is smaller than the Seifert genus of $K$, then $K$ has Property G. When
the smooth $4$--genus is $0$, $Y$ can be taken to be any closed, oriented
$3$--manifold.
