We introduce the notion of domain of finite type \(\mathscr {D}\subset {\mathbb {R}}^n\) generalizing an earlier work of Bodin, Popescu-Pampu and Sorea. Then, we prove that every finite graph admitting a good orientation whose vertices have degree 1 or 3 can be realized as the Poincaré-Reeb graph of a stable (globally) algebraic domain of finite type \(\mathscr {D}\subset {\mathbb {R}}^n\) , for every \(n\ge 2\) . If in addition \(n\ge 3\) , we construct a class of graphs allowing vertices of degree 2 also. Algebraic approximation techniques á la Nash-Tognoli and stable Morse functions are fundamental tools in our approach. In particular, the recent extensions over \({\mathbb {Q}}\) of such algebraic approximation techniques developed by Ghiloni and the author allow us to reduce the coefficients of the describing polynomials over \({\mathbb {Q}}\) and to extend our constructions over real closed fields.
