The integral cohomology rings of four-dimensional toric orbifolds

Let $X(P,\lambda )$ be a 4-dimensional toric orbifold associated to a polygon $P$ and a characteristic function $\lambda$. Assuming that $X(P,\lambda )$ is locally smooth over a vertex of $P$, we determine the integral cohomology ring $H^{\ast }(X(P,\lambda );\mathbb{Z})$ by constructing an explicit basis and expressing the cup products of the basis elements in terms of $P$ and $\lambda$.