Inquiries such as whether a task A depends on a task B, whether an author A has been influenced by a paper B, whether a certain protein is associated with a specific biological process or molecular function, or whether class A inherits from class B, are just a few examples of inquiries that can be modeled as reachability queries on a network (Directed Graph). Digital systems answer myriad such inquiries every day. In this paper, we discuss the transitive closure problem. We focus on applicable solutions that enable us to answer queries fast, in constant time, and can serve in real-world applications. In contrast to the majority of research on this topic that revolves around the construction of a two-dimensional adjacency matrix, we present an approach that builds a reachability indexing scheme. This scheme enables us to answer queries in constant time and can be built in parameterized linear time. In addition, it captures a compressed data structure. Our approach and algorithms are validated by extensive experiments that shed light on the factors that play a key role in this problem. To stress the efficiency of this solution and demonstrate the potential to apply our approach to important problems, we use it to speed up Fulkerson's method for finding the width of a DAG. Our results challenge the prevailing belief, reiterated over the last thirty years, regarding the efficiency of this method. Our approach is based on the concept of chain decomposition. Before we delve into its description, we introduce, analyze, and utilize a chain decomposition algorithm. Furthermore, we explore how chain decomposition can facilitate transitive closure solutions introducing a general purpose linear time reduction technique that removes a large subset of transitive edges given any chain decomposition.