  For given closed orientable 3-manifolds $M$ and $N$ let $\c{D}(M,N)$ be the
set of mapping degrees from $M$ to $N$. We address the problem: For which $N$,
$\c{D}(M,N)$ is finite for all $M$? The answer is known in Thurston's picture
of closed orientable irreducible 3-manifolds unless the target is a non-trivial
graph manifold. We prove that for each closed non-trivial graph manifold $N$,
$\c{D}(M,N)$ is finite for all graph manifold $M$.
  The proof uses a recently developed standard forms of maps between graph
manifolds and the estimation of the $\widetilde{\rm PSL}(2,{\R})$-volume for
certain class of graph manifolds.
