Definitions Directed acyclic graph

adding red edges blue directed acyclic graph produces dag, transitive closure of blue graph. each red or blue edge uv, v reachable u: there exists blue path starting @ u , ending @ v.

a vertex v of directed graph said reachable vertex u when there exists path starts @ u , ends @ v. special case, every vertex considered reachable (by path 0 edges). if vertex can reach via nontrivial path (a path 1 or more edges), path cycle, way define directed acyclic graphs graphs in no vertex can reach via nontrivial path.

a topological ordering of directed graph ordering of vertices sequence, such every edge start vertex of edge occurs earlier in sequence ending vertex of edge. graph has topological ordering cannot have cycles, because edge earliest vertex of cycle have oriented wrong way. therefore, every graph topological ordering acyclic.

conversely, every directed acyclic graph has topological ordering. therefore, property can used alternative definition of directed acyclic graphs: graphs have topological orderings.