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The transitive reduction of a finite directed graph G is a graph with the fewest possible edges that has the same reachability relation as the original graph. That is, if there is a path from a vertex x to a vertex y in graph G, there must also be a path from x to y in the transitive reduction of G, and vice versa.
The transitive reduction of a DAG is the graph with the fewest edges that has the same reachability relation as the DAG. It has an edge u → v for every pair of vertices ( u , v ) in the covering relation of the reachability relation ≤ of the DAG.
Specifically, taking a strict partial order relation (, <), a directed acyclic graph (DAG) may be constructed by taking each element of to be a node and each element of < to be an edge. The transitive reduction of this DAG [b] is then the Hasse diagram. Similarly this process can be reversed to construct strict partial orders from certain DAGs.
The transitive closure of this relation is a different relation, namely "there is a sequence of direct flights that begins at city x and ends at city y". Every relation can be extended in a similar way to a transitive relation. An example of a non-transitive relation with a less meaningful transitive closure is "x is the day of the week after y".
I want them soft, creamy, not at all dry, and quite possibly a bit less "done" than some people like them. My go-to method for years has been (for two of us): five large eggs, one large yolk, salt ...
The Southern living test kitchen always uses this butter for baking. Food. Food & Wine. Even pro chefs use instant ramen — here’s how they make it restaurant-quality. Lighter Side.
You seem to be looking at the figures with captions "A DAG" and "its transitive reduction". The transitive reduction is not the same thing as the transitive closure. —David Eppstein 02:13, 6 December 2024 (UTC) Ahh! Of course. I should read more carefully. Thanks for the prompt explanation. LachlanA 02:19, 6 December 2024 (UTC)
To test that theory, USA TODAY reached out to retirees across the country who are living mostly on Social Security and asked how they were doing.