# Definition:Transitive Reduction

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## Definition

### Relation Theory

Let $\RR$ be a relation on a set $S$.

A **transitive reduction** of $\RR$ is denoted $\RR^-$, and is defined as a minimal relation on $S$ which has the same transitive closure as $\RR$.

### Graph Theory

The concept of transitive reduction is usually encountered in the field of graph theory where it has considerable importance:

Let $G = \struct {V, E}$ be a loop-digraph.

Let $G$ be expressed formally as a relational structure $\GG$.

A **transitive reduction** of $G$ is denoted $G^-$, and is defined as a transitive reduction of the relation $\GG$.

Hence it is a minimal loop-digraph on $V$ which has the same transitive closure as $\GG$.

## Also see

- Results about
**transitive reductions**can be found here.