 Acyclic orientation

An acyclic orientation is an orientation of a graph whose resulting directed graph has no directed cycle, hence is a Directed acyclic graph. A directed cycle must follow every arrow around its closed route. A cycle in a graph in the skeleton of a directed graph only requires the corresponding undirected edges; their arrows need not agree around the route.

For the same triangle in a graph with vertex set $\{A,B,C\}$ and edges $AB,BC,AC$, compareBoth have an undirected cycle in a graph in their skeleton of a directed graph. Only $D_2$ has a directed cycle. In $D_1$, $C$ has no outgoing directed edge, so a route cannot continue from $C$ back to $A$.

Thus an acyclic orientation does not require its underlying undirected graph to be a forest. Every orientation of a graph of a forest is acyclic, because a directed cycle would give an undirected cycle in a graph. Conversely, every finite simple graph, including one with many cycles in a graph, admits at least one acyclic orientation by the following ordering construction. On this triangle in a graph, six of the eight orientations of a graph are acyclic; the other two follow the directed cycle in opposite senses.

An acyclic orientation of a directed graph (digraph) is an assignment of directions to the edges of the graph such that there are no directed cycles.