According to [Wikipedia](https://en.wikipedia.org/wiki/Transitive_reduction#CITEREFAhoGareyUllman1972), *"If a given graph is a finite directed acyclic graph, its transitive reduction is unique"*


Here is what I think might be a counter-example:


Imagine a diamond-shaped DAG where 
A -> B, A -> C, B -> D, C -> D


I am thinking there could be two different transitive reductions: 


1) A -> B, A -> C, B -> D


and 


2) A -> B, A -> C, C -> D


Where am I mistaken?



Is transitive reduction for a direct acyclic graph really unique?
In the first reduction you mention, you are removing C -> D, and so your subgraph does not mention that D is reachable from C. So your subgraph is not a transitive reduction. As the Wikipedia article mentions, the transitively reduced digraph must have the same reachability relation as the given relation. The original graph is already transitively reduced. 


If the original graph also had the arc A -> D, then this arc can be removed while reducing. 


