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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.
[8] [9] For instance, for W = 3, this means that it uses at most 4/3 times as many levels as is optimal. When the partial order of precedence constraints is an interval order, or belongs to several related classes of partial orders, the Coffman–Graham algorithm finds a solution with the minimum number of levels regardless of its width bound. [10]
The data structure is typically stored as a Boolean matrix, so if matrix[1][4] = true, then it is the case that node 1 can reach node 4 through one or more hops. 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.
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 correct evaluation order is a numbering : of the objects that form the nodes of the dependency graph so that the following equation holds: () < (,) with ,. This means, if the numbering orders two elements a {\displaystyle a} and b {\displaystyle b} so that a {\displaystyle a} will be evaluated before b {\displaystyle b} , then a ...
The butterfly network, a multitree used in distributed computation, showing in red the undirected tree induced by the subgraph reachable from one of its vertices.. In combinatorics and order theory, a multitree may describe either of two equivalent structures: a directed acyclic graph (DAG) in which there is at most one directed path between any two vertices, or equivalently in which the ...
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.