Recently, I read definition of Speiser graph or, also called, line complex (see, for example there ). There is a certain ambiguity in its definition and I am going to formulate my question about it. Just in case, after my question I provide all necessary notions.
Let $f\colon \mathbb{C} \to \mathbb{C}$ be an entire map of finite type. If we pick some simple closed curve $L$ passing through singular set $S(f)$ of the map $f$, we can construct Speiser graph $\Gamma_{f, L}$ of $f$. However, there are different choices of curve $L$ and one can construct examples graphs $\Gamma_{f, L_1}$ and $\Gamma_{f, L_2}$ that are different for different $L_1$ and $L_2$. Here by equal graphs I mean graphs which can be mapped to each other by homeomorphism of $\mathbb{C}$ which sends vertices of one graph to vertices of another.
In principle, there are infinitely many possible curves $L$ to pick, but is there are any map for which we can construct infinitely many different Speiser graphs $\Gamma_{f, L}$ starting from different curves $L$?
For instance, in the case when $f$ is polynomial, any Speiser graph is always finite, i.e., we have only finitely many possibilities. One can easily prove that it is also true for some entire maps, for example, $\exp(z)$, $\sin(z)$, $\cos(z)$, $p(\exp(z))$, where $p$ is a polynomial. But in general, the answer is not clear for me.
Notions:
Let $X$ be a connected and simply connected topological surface, $f\colon X \to \widehat{\mathbb{C}}$ be a continuous, open and discret (i.e., $f^{-1}(z)$ is discrete for any $z \in \widehat{\mathbb{C}}$) map. Suppose that minimal set $S(f)$, such that $f\colon X \setminus f^{-1}(S(f)) \to \widehat{\mathbb{C}}\setminus S(f)$ is a covering, is finite. Here set $S(f)$ is called singular set of $f$ and $f$ is called map of finite type. For example, any polynomial $p$ is entire map of finite type, as well as a lot of other maps, like $\exp(z), \sin(z), \cos(z), p(\exp(z)), \exp(p(z))$, etc. Entire map $\cos(z) + z$ is an easy examples of non-finite type map, because it has infinitely many critical values.
Let $S(f) = \{a_1, a_2, \dots, a_q\}$. Pick any continuous simple closed curve $L$ in $\widehat{\mathbb{C}}$ passing through points of $S(f)$. Let us assume that $L$ passes through points of $S(f)$ in the order of their indices. This curve consists of $q$ arcs $l_i$, where $l_i$ has endpoints $a_{i - 1}$ and $a_i$. Here and further indices considered modulo $q$.
Complement of $L$, i.e., $\widehat{\mathbb{C}} \setminus L$, consists of two regions $H_1$ and $H_2$. Pick two points $p_1 \in H_1$ and $p_2 \in H_2$ and construct $q$ Jordan arcs $\alpha_i$ joining $p_1$ and $p_2$ such that $\alpha_i$ intersects $L$ atexactly one point located on $l_i \setminus S(f)$. Now consider planar embedded graph $\Gamma_L$ with vertices $\{p_1, p_2\}$ and edge set $\alpha_i, i = 1, 2, \dots, q$. In other words, graph $\Gamma_L$ is dual to the curve $L$ viewed as graph with vertex set $S(f)$.
Now consider planar embedded graph $\Gamma_{f, L} := f^{-1}(\Gamma_L)$ as a graph with vertex set $f^{-1}(S(f))$. It happens to be sufficiently nice graph, for instance, it is connected, regular (i.e., degree of every vertex equals to $q$), bipartite, and it covers graph $\Gamma$ by the map $f|f^{-1}(\Gamma)$.
This graph is called a Speiser graph or line complex of the map $f$. Note that for a given finite type map $f\colon X \to \widehat{\mathbb{C}}$ it is defined up to choice of curve $L$.