  For each $k\in\mathbb{A}^4(\mathbb{C})$, consider the character variety $X_k$
on a four-holed sphere. We prove that it is decidable whether or not any two
integral solutions of $X_k$ are in the same mapping class group orbit. For
this, using a delta map and Vieta maps, we will introduce graphs corresponding
to the orbits and observe the properties of vertices on the graph.
