directedtransitivity

Directed transitivity is a property of a directed graph or, equivalently, a binary relation on a set. A directed graph is transitive if, whenever there is a directed edge from a to b and a directed edge from b to c, there is also a directed edge from a to c. In relation terms, for all elements x, y, z, if xRy and yRz then xRz.

Self-loops are not required by transitivity; a graph can be transitive without containing any edge from a vertex to itself. A transitive graph may still contain cycles, and the presence of cycles interacts with transitivity in ways that can affect the interpretation of reachability and closure.

Transitive closure is a related concept: it is the smallest transitive graph that contains the original edge set. Equivalently, it adds exactly those edges needed to ensure the transitivity condition holds for all triples. In a directed graph, the transitive closure corresponds to reachability: an edge a→b is in the transitive closure if and only if there exists a directed path from a to b in the original graph.

Examples illustrate the idea. If a→b and b→c are present but a→c is missing, the graph is not transitive. Adding a→c yields a transitive graph. If, in addition, a→d and c→d exist with b→d, the same principle applies to ensure all required edges are present.

Algorithms for computing transitive closure include Warshall’s algorithm and, more generally, performing reachability searches from each vertex. The closure is used in databases, knowledge graphs, program analysis, and formal reasoning to determine indirect connections and to optimize query processing and type checks. In summary, directed transitivity describes how local edge patterns enforce longer-range connections in directed structures.