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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 $z$.

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.

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 $z$.

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 $z$.

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 $z$.

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 $z$.