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 the following is true.

>**Claim.** Any such $\Gamma$ is isometric to the space obtained by gluing few copies of $[0,\pi]$ along the ends.

Assume the later is proved.
Let $f\colon \Gamma\to\mathbb S^n$ be 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 $\Gamma$ lies in the half-sphere with center $z$.