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.
Assume the later is proved. Take the images of the ends, say $x,y\in \mathbb S^n$. Let $z$ be the midpoint of the minimizing geodesic from $x$ and $y$ in $\mathbb S^n$. Then the image of each arc and therefore the image of whole $\Gamma$ lies in the half-sphere with center $z$.