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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 closure of a DAG is the graph with the most 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 reachability relation ≤ of the DAG, and may therefore be thought of as a direct translation of the reachability relation ≤ into graph-theoretic terms.
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".
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.
A Hasse diagram of the factors of 60 ordered by the is-a-divisor-of relation. In order theory, a Hasse diagram (/ ˈ h æ s ə /; German:) is a type of mathematical diagram used to represent a finite partially ordered set, in the form of a drawing of its transitive reduction.
A top Federal Reserve official said Monday that he is leaning toward supporting an interest rate cut when the Fed meets in two weeks but that evidence of persistent inflation before then could ...
One of the most common examples is the reduction of a general directed graph to an acyclic directed graph by contracting all of the vertices in each strongly connected component. If the relation described by the graph is transitive , no information is lost as long as we label each vertex with the set of labels of the vertices that were ...