Let $(X,0)$ be a complex surface germ with an isolated singularity and $I$ be an $\mathfrak{m}\text{-primary}$ ideal (contains a power of the maximal ideal $\mathfrak{m}$) of the local ring $\mathcal{O}_{X,0}$. The minimal log resolution of $I$ is the minimal good resolution $\pi_I:(X_I,E) \longrightarrow (X,0)$ such that the ideal sheaf $\pi^*I$ is locally principal. Denote by $\Gamma_I$ the dual graph associated to the resolution $\pi_I$.
My question is the following: Let $I$ and $J$ be two $\mathfrak{m}$-primary ideals of $\mathcal{O}_{X,0}$. is there a condition on $I$ and $J$ such that the graph $\Gamma_I=\Gamma_J$. I mean by that the equality of graph, for example: $(X,0)=(\mathbb{C}^2,0)$ and consider the ideals $\mathfrak{m}^n$ only one blow up $b:(\mathrm{BL}_0(\mathbb{C}^2),E)\longrightarrow (\mathbb{C}^2,0)$ makes the sheaves $b^*\mathfrak{m}^n$ locally principal, then $b$ is the minimal log resolution of the ideals $\mathfrak{m}^n$ and the associated dual graph is just one vertex which means $\Gamma_{\mathfrak{m}^n}=\Gamma_{\mathfrak{m}^m}$. On the other hand i take the family of ideals $(x^P,y^q)$ with $p$ and $q$ coprime the graph are different.
Thanks in advance for your help.
