Poincaré inequalities for maps with target manifold of negative curvature

We prove that for any given homotopic $C^1$-maps $u,v\colon G\to M$ in a nontrivial homotopy class from a metric graph into a closed manifold of negative sectional curvature, the distance between $u$ and $v$ can be bounded by $3({\rm length}(u)+{\rm length}(v))+C(\kappa,\varrho/20)$, where $\varrho>0$ is a lower bound of the injectivity radius and $-\kappa<0$ an upper bound for the sectional curvature of $M$. The constant $C(\kappa,\varepsilon)$ is given by
$$ C(\kappa,\varepsilon)=8\sh_\kappa^{-1}(1)+8\sh_\kappa^{-1}(\varepsilon)) $$
with $\sh_\kappa(t)=\sinh(\sqrt{\kappa}t)$. Various applications are given.