On the minimal free resolution of the Rees algebra of tri-generated bivariate monomial ideals

Let $I$ be a monomial ideal in two variables generated by three monomials and let $\mathcal{R}(I)$ be its Rees ideal. We describe an algorithm to compute the minimal generating set of $\mathcal{R}(I)$. Based on the data obtained by this algorithm, we build a graph that encodes the minimal free resolution of $\mathcal{R}(I)$. We explicitly describe the modules and differentials on the minimal free resolution of $\mathcal{R}(I)$.