o $L_n \subset {\mathbb{R}^2}$. Suppose that for a transformation ${\phi}_{n}$ on $L_n$ defined as $${\phi}_{n}(r \cos \frac{2k\pi}{n}, r \sin \frac{2k\pi}{n})=(r \cos \frac{2(k+1)\pi}{n}, r \sin \frac{2(k+1)\pi}{n})$$ there exists a $C^s$ diffeomorphism ${\Phi}_{n}$ satisfying ${\phi}_n \circ q=q \circ {\Phi}_n$ and that for one ${\phi}_{r}$ on $L_n$ defined as $${\phi}_{r}(r \cos \frac{2k\pi}{n}, r \sin \frac{2k\pi}{n})=(r \cos \frac{-2k\pi}{n}, r \sin \frac{-2k\pi}{n})$$ there exists a $C^s$ diffeomorphism ${\Phi}_{r}$ satisfying ${\phi}_r \circ q=q \circ {\Phi}_r$. Then $q$ is said to be [*$D_n$-symmetric*]{}.

\[thm:1\] There exist a class $\mathcal{C}$ of $(C^r,C^0)$ maps whose Reeb spaces are regarded as Reeb graphs and a class ${\mathcal{Q}}_{\mathcal{C}}$ of pseudo quotient maps of the class satisfying the following.

1.  For maps of the class $\mathcal{C}$, inverse images of regular values are disjoint unions of standard spheres.

2.  For maps of the class ${\mathcal{Q}}_{\mathcal{C}}$, around each vertex of degree $n>1$, the map is regarded as a $D_n$-symmetric map onto $L_n$ by applying a suitable identification.

3.  For any finite connected graph which is not a single point, we can construct a map of the class ${\mathcal{Q}}_{\mathcal{C}}$ onto the graph.

4.  For a map of the class ${\mathcal{Q}}_{\mathcal{C}}$, if for the target graph, for each vertex, the degree is at most $3$, then the map is realized as a map into $S^1$ or $\mathbb{R}$ of the class $\mathcal{C}$.

In the proof, we first perform local construction around each vertex (Step 1–3) and in Step 4, we complete the construction by constructing remaining parts. Last, we give the strict definitions of $\mathcal{C}$ and $\mathcal{Q}_{\mathcal{C}}$. We will see that this completes the proof except for the fourth condition. Last we discuss the fourth condition.

Step 1 Around a vertex of degree $2$.\
We consider a trivial $C^r$ bundle over $[-1,1]$ whose fiber is a standard sphere. We compose a surjective function over $[-1,1]$ defined by 