Editing Directed acyclic graph (section)

=== Construction from cyclic graphs ===
Any undirected graph may be made into a DAG by choosing a [[total order]] for its vertices and directing every edge from the earlier endpoint in the order to the later endpoint. The resulting [[Orientation (graph theory)|orientation]] of the edges is called an [[acyclic orientation]]. Different total orders may lead to the same acyclic orientation, so an {{mvar|n}}-vertex graph can have fewer than {{math|''n''!}} acyclic orientations. The number of acyclic orientations is equal to {{math|{{!}}''χ''(−1){{!}}}}, where {{mvar|χ}} is the [[chromatic polynomial]] of the given graph.<ref>{{citation|first=Richard P.|last=Stanley|author-link=Richard P. Stanley|title=Acyclic orientations of graphs|journal=Discrete Mathematics|volume=5|issue=2 |pages=171–178|year= 1973|doi=10.1016/0012-365X(73)90108-8|url=http://math.mit.edu/~rstan/pubs/pubfiles/18.pdf}}.</ref>

[[File:Graph Condensation.svg|thumb|upright=1.5|The yellow directed acyclic graph is the [[Condensation (graph theory)|condensation]] of the blue directed graph. It is formed by [[Edge contraction|contracting]] each [[strongly connected component]] of the blue graph into a single yellow vertex.]]
Any directed graph may be made into a DAG by removing a [[feedback vertex set]] or a [[feedback arc set]], a set of vertices or edges (respectively) that touches all cycles. However, the smallest such set is [[NP-hard]] to find.<ref>{{Garey-Johnson}}, Problems GT7 and GT8, pp.&nbsp;191–192.</ref> An arbitrary directed graph may also be transformed into a DAG, called its [[condensation (graph theory)|condensation]], by [[Edge contraction|contracting]] each of its [[strongly connected component]]s into a single supervertex.<ref>{{citation|title=Structural Models: An Introduction to the Theory of Directed Graphs|last1=Harary|first1=Frank|author1-link=Frank Harary|last2=Norman|first2=Robert Z.|last3=Cartwright|first3=Dorwin|publisher=John Wiley & Sons|year=1965|page=63}}.</ref> When the graph is already acyclic, its smallest feedback vertex sets and feedback arc sets are [[empty set|empty]], and its condensation is the graph itself.