  In this article, we introduce a generalization of the concept of graded
$r$-ideals in graded commutative rings with nonzero unity. Let $G$ be a group,
$R$ be a $G$-graded commutative ring with nonzero unity and $GI(R)$ be the set
of all graded ideals of $R$. Suppose that $\phi: GI(R)\rightarrow
GI(R)\bigcup\{\emptyset\}$ is a function. A proper graded ideal $P$ of $R$ is
called a graded $\phi$-$r$-ideal of $R$ if whenever $x, y$ are homogeneous
elements of $R$ such that $xy\in P-\phi(P)$ and $Ann(x) =\{0\}$, then $y\in P$.
Several properties of graded $\phi$-$r$-ideals have been examined.
