Order-induced acyclic orientation

Choose a strict total order $v_1<\cdots <v_n$ of the vertices of a finite simple graph. Direct every edge $v_i{v}_j$ from $v_i$ to $v_j$ when $i<j$. Along every directed path, the index strictly increases. Returning to the starting vertex would require an impossible strict increase back to the initial index, so this is an acyclic orientation. The displayed $D_1$ comes from $A<B<C$.

The chosen strict total order is a topological order of the resulting Directed acyclic graph. Conversely, every acyclic orientation of a finite simple graph has a topological order, and orienting its edges from earlier to later in that order recovers it. Different topological orders can induce the same acyclic orientation, since vertices with no edge between them need not impose a choice of arrow.^[ref]

An order respects reachability but does not create it: $u<v$ in a chosen topological order does not imply a directed path from $u$ to $v$. For example, orient the two edges of the graph path $A,B,C$ using $A<C<B$. The arrows are $A\to B$ and $C\to B$; there is no directed path from $A$ to $C$, despite $A<C$. In $D_1$, however, $A$ reaches $C$ both directly and via $B$. Following outgoing directed edges with graph search decides this directional reachability.^[ref]