For example, a DAG with two edges u → v and v → w has the same reachability relation as the DAG with three edges u → v, v → w, and u → w. However, different DAGs may give rise to the same reachability relation and the same partial order.
In this partial order, two vertices u and v are ordered as u ≤ v exactly when there exists a directed path from u to v in the DAG that is, when u can reach v (or v is reachable from u). The reachability relation of a DAG can be formalized as a partial order ≤ on the vertices of the DAG. Mathematical properties Reachability relation, transitive closure, and transitive reduction If a vertex can reach itself via a nontrivial path (a path with one or more edges), then that path is a cycle, so another way to define directed acyclic graphs is that they are the graphs in which no vertex can reach itself via a nontrivial path. As a special case, every vertex is considered to be reachable from itself (by a path with zero edges). Ī vertex v of a directed graph is said to be reachable from another vertex u when there exists a path that starts at u and ends at v. A directed acyclic graph is a directed graph that has no cycles. A path in a directed graph is a sequence of edges having the property that the ending vertex of each edge in the sequence is the same as the starting vertex of the next edge in the sequence a path forms a cycle if the starting vertex of its first edge equals the ending vertex of its last edge. In the case of a directed graph, each edge has an orientation, from one vertex to another vertex. Definitions Ī graph is formed by vertices and by edges connecting pairs of vertices, where the vertices can be any kind of object that is connected in pairs by edges. DAGs have numerous scientific and computational applications, ranging from biology (evolution, family trees, epidemiology) to information science (citation networks) to computation (scheduling).ĭirected acyclic graphs are sometimes instead called acyclic directed graphs or acyclic digraphs. A directed graph is a DAG if and only if it can be topologically ordered, by arranging the vertices as a linear ordering that is consistent with all edge directions. That is, it consists of vertices and edges (also called arcs), with each edge directed from one vertex to another, such that following those directions will never form a closed loop. In mathematics, particularly graph theory, and computer science, a directed acyclic graph ( DAG) is a directed graph with no directed cycles.
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