Let $\Gamma$ be the metric space glued from segments of length $\pi/n$ along the rule described by your graph. 

Note that $\Gamma$ is a 1-dimensional spherical building.
Therefore your question can be reformulated the following way:

>Let $\Gamma$ be a 1-dimensional spherical building and $f\colon \Gamma\to\mathbb S^n$ is a contracting map. Then the image $f(\Gamma)$ lies in a half-sphere.