On the structure of $S_2$-ifications of complete local rings

Motivated by work of Hochster and Huneke, we investigate several constructions related to the $S_2$-ification $T$ of a complete equidimensional local ring $R$: the canonical module, the top local cohomology module, topological spaces of the form $\operatorname{Spec}(R)-V(J)$, and the (finite simple) graph $\Gamma_R$ with vertex set $\operatorname{Min}(R)$ defined by Hochster and Huneke. We generalize one of their results by showing, e.g., that the number of maximal ideals of $T$ is equal to the number of connected components of $\Gamma_R$. We further investigate this graph by exhibiting a technique for showing that a given graph $G$ can be realized as one of the form $\Gamma_R$.