# transitive reduction

(definition)

**Definition:**
The transitive reduction of a *directed graph* G is the directed graph G' with the smallest number of *edges* such that for every *path* between *vertices* in G, G' has a path between those vertices.

**See also**
*reduced digraph*, *transitive closure*.

*Note:
Informally G' is the minimal graph with the same connectivity as G. After abstract of ***A. V. Aho, M. R. Garey**, and **J. D. Ullman**. *The transitive reduction of a directed graph*. SIAM Journal on Computing, 1:131--137, 1972.

Author: PEB

## Implementation

on a graph (C++ and Mathematica)

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Paul E. Black.

Entry modified 3 November 2010.

HTML page formatted Tue Dec 6 16:16:33 2011.

Cite this as:

Paul E. Black, "transitive reduction", in
*Dictionary of Algorithms and Data
Structures* [online], Paul E. Black, ed.,
U.S. National Institute of
Standards and Technology. 3 November 2010. (accessed TODAY)
Available from: http://www.nist.gov/dads/HTML/transitiveReduction.html