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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 the adjacency relation of a directed acyclic graph (DAG) is the reachability relation of the DAG and a strict partial order. A cluster graph, the transitive closure of an undirected graph. The transitive closure of an undirected graph produces a cluster graph, a disjoint union of cliques.
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 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.
U.S. farm industry groups want President-elect Donald Trump to spare their sector from his promise of mass deportations, which could upend a food supply chain heavily dependent on immigrants in ...
A depends on B and C; B depends on D. Given a set of objects and a transitive relation with (,) modeling a dependency "a depends on b" ("a needs b evaluated first"), the dependency graph is a graph = (,) with the transitive reduction of R.