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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.
If is acyclic, then its reachability relation is a partial order; any partial order may be defined in this way, for instance as the reachability relation of its transitive reduction. [2] A noteworthy consequence of this is that since partial orders are anti-symmetric, if s {\displaystyle s} can reach t {\displaystyle t} , then we know that t ...
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".
The transitive reduction or covering graph of the Dedekind–MacNeille completion describes the order relation between its elements in a concise way: each neighbor of a cut must remove an element of the original partial order from either the upper or lower set of the cut, so each vertex has at most n neighbors.
Elon Musk's Department of Government Efficiency has sparked concerns within the intelligence community after it posted information about an agency that oversees U.S. intelligence satellites to its ...
A Hasse diagram is a type of mathematical diagram used to represent a finite partially ordered set, in the form of a drawing of its transitive reduction. Homogeneous relation. A homogeneous relation on a set is a subset of . Said differently, it is a binary relation over and itself.