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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.
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 ...
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 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 extension of R 1 would be denoted by R 2, and continuing in this way, in general, the transitive extension of R i would be R i + 1. The transitive closure of R, denoted by R* or R ∞ is the set union of R, R 1, R 2, ... . [8] The transitive closure of a relation is a transitive relation. [8]
closure 1. For the transitive closure of a directed graph, see transitive. 2. A closure of a directed graph is a set of vertices that have no outgoing edges to vertices outside the closure. For instance, a sink is a one-vertex closure. The closure problem is the problem of finding a closure of minimum or maximum weight. co-
A transitive orientation of a graph is an acyclic orientation that equals its own transitive closure. Not every graph has a transitive orientation; the graphs that do are the comparability graphs. [8] Complete graphs are special cases of comparability graphs, and transitive tournaments are special cases of transitive orientations.