Description of some ground states by Puiseux technics

Let $(\Sigma^+_G, \sigma)$ be a one-sided transitive subshift of finite type, where symbols are given by a finite spin set $ S $, and admissible transitions are represented by an irreducible directed graph $ G\subset S\times S $. Let $ H : \Sigma^+_G\to\mathbb{R}$ be a locally constant function (that corresponds with a local observable which makes finite-range interactions). Given $\beta > 0$, let $ \mu_{\beta H} $ be the Gibbs-equilibrium probability measure associated with the observable $-\beta H$. It is known, by using abstract considerations, that $\{\mu_{\beta H}\}_{\beta>0}$ converges as $ \beta \to + \infty $ to a $H$-minimizing probability measure $\mu_{\textrm{min}}^H$ called zero-temperature Gibbs measure. For weighted graphs with a small number of vertices, we describe here an algorithm (similar to the Puiseux algorithm) that gives the explicit form of $\mu_{\textrm{min}}^H$ on the set of ground-state configuration