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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.
The Floyd–Warshall algorithm [5] can be used to compute the transitive closure of any directed graph, which gives rise to the reachability relation as in the definition, above. The algorithm requires (| |) time and (| |) space in the worst case. This algorithm is not solely interested in reachability as it also computes the shortest path ...
The Floyd–Warshall algorithm is an example of dynamic programming, and was published in its currently recognized form by Robert Floyd in 1962. [3] However, it is essentially the same as algorithms previously published by Bernard Roy in 1959 [4] and also by Stephen Warshall in 1962 [5] for finding the transitive closure of a graph, [6] and is closely related to Kleene's algorithm (published ...
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.
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-
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This equivalence relation is a transitive closure of the relation defined by (=) (=), which highlights the fact that a compacted automaton may be obtained by both gluing suffix tree vertices equivalent via = relation (minimization of the suffix tree) and gluing suffix automaton states equivalent via = relation (compaction of suffix automaton). [23]