Post Undeleted by Anton Petrunin
4 deleted 492 characters in body

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 diameter of $\Gamma$ is $\pi$ and any two points on distance $<\pi$ are joint by unique geodesic. (BTW this means that $\Gamma$ is a $CAT(1)$ space of diameter $\pi$ with extendable geodesics.)

It seems that 1-dimensional spherical building. Therefore your question can be reformulated the following is true.

Claim. Any such way:

Let $\Gamma$ is isometric to the space obtained by gluing few copies of $[0,\pi]$ along the ends.

Assume the later is proved. Let be a 1-dimensional spherical building and $f\colon \Gamma\to\mathbb S^n$ be is a contracting map. 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 $\Gamma$ f(\Gamma)$lies in the a half-spherewith center$z\$..

Post Deleted by Anton Petrunin
3 added 247 characters in body
2 added 34 characters in body
1