The chase is a well-established family of algorithms used to materialize Knowledge Bases (KBs), like Knowledge Graphs (KGs), to tackle important tasks like query answering under dependencies or data cleaning. A general problem of chase algorithms is that they might perform redundant computations. To counter this problem, we introduce the notion of Trigger Graphs (TGs), which guide the execution of the rules avoiding redundant computations. We present the results of an extensive theoretical and empirical study that seeks to answer when and how TGs can be computed and what are the benefits of TGs when applied over real-world KBs. Our results include introducing algorithms that compute (minimal) TGs. We implemented our approach in a new engine, and our experiments show that it can be significantly more efficient than the chase enabling us to materialize KBs with 17B facts in less than 40 min on commodity machines.