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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.
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 ...
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 closure of this relation is a different relation, namely "there is a sequence of direct flights that begins at city x and ends at city y". Every relation can be extended in a similar way to a transitive relation. An example of a non-transitive relation with a less meaningful transitive closure is "x is the day of the week after y".
A Hasse diagram of the factors of 60 ordered by the is-a-divisor-of relation. In order theory, a Hasse diagram (/ ˈ h æ s ə /; German:) is a type of mathematical diagram used to represent a finite partially ordered set, in the form of a drawing of its transitive reduction.
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.
Pourbaix diagram of iron. [1] The Y axis corresponds to voltage potential. In electrochemistry, and more generally in solution chemistry, a Pourbaix diagram, also known as a potential/pH diagram, E H –pH diagram or a pE/pH diagram, is a plot of possible thermodynamically stable phases (i.e., at chemical equilibrium) of an aqueous electrochemical system.