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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.
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.
The figures with the captions "A DAG" and "its transitive closure" seem to be the wrong way around. I would fix it myself, except the figures are names Tred-G.svg and Tred-Gprime.svg, which suggests that the one which I think is the transitive closure is really intended to be the original, and it isn't just that the figures were accidentally in ...
The relation is defined as the transitive closure of . That is, u ≍ v {\displaystyle u\asymp v} when there is a sequence u ≈ ⋯ ≈ v {\displaystyle u\approx \cdots \approx v} of vertices, starting with u {\displaystyle u} and ending with v {\displaystyle v} , such that each consecutive pair in the sequence is related by ≈ {\displaystyle ...
A transitive orientation is an orientation such that the resulting directed graph is its own transitive closure. The graphs with transitive orientations are called comparability graphs; they may be defined from a partially ordered set by making two elements adjacent whenever they are comparable in the partial order. [8] A transitive orientation ...
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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]