First let me reformulate the problem in a more geometric way.

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 $CAT(1)$ space of diameter $\pi$ with extendable geodesics.
It seems that any such $\Gamma$ is obtained by gluing few copies of $[0,\pi]$ along the ends.

Once the later is proved, take the images of the ends, say $x$ and $y$ in the $\mathbb S^n$
and let $z$ be the midpoint of the arc $[xy]$. Then the image of each arc and therefore the image of whole $\Gamma$ lies in the half-sphere with center $z$.