Strong orientation

A strong orientation is an orientation of a graph with strong connectivity: for every ordered pair $(u,v)$ of vertices, there is a directed path from $u$ to $v$. A length-zero directed path permits $u=v$. Ordinary connectivity of the underlying undirected graph is necessary, but does not ensure this two-way reachability after directions are chosen. The acyclic $D_1$ is not a strong orientation; the cyclic $D_2$ is, since a journey can continue around its directed cycle.

Attach a new vertex $D$ to that triangle in a graph by the single edge $CD$. This edge is a bridge in a graph. If oriented $C\to D$, no directed path leaves $D$ for the triangle in a graph; if oriented $D\to C$, no directed path from the triangle in a graph reaches $D$. Neither choice gives a strong orientation, even when the triangle in a graph itself is cyclically oriented. By contrast, the four-vertex cycle graph has the strong orientationFollowing this directed cycle reaches any requested vertex in at most three steps. Removing any one of its undirected edges leaves a graph path through all four vertices, so it has no bridge in a graph.

For more than one vertex, a strong orientation cannot also be an acyclic orientation: directed paths from $u$ to $v$ and back give a nonempty closed directed walk, from which removing repeated portions leaves a directed cycle. A directed graph containing one directed cycle need not have strong connectivity, as the attached $D$ shows.

A finite nonempty connected graph whose edges are undirected admits a strong orientation if and only if it has no bridge in a graph. This statement holds for finite multigraphs as well as finite simple graphs; self-loops do not help connect distinct vertices. The single-vertex case is allowed, with length-zero directed paths.^[ref]

Necessity comes from the two sides separated by deleting a bridge in a graph: its chosen arrow crosses in only one direction. For sufficiency, the cited constructive proof uses depth-first search to find a cycle in a graph, orients it cyclically, contracts it, and repeats. Undoing the contractions replaces junctions by directed cycles, retaining two-way reachability. Contraction can produce multiple edges, so these must remain distinct. This is a construction outline; the input graph is finite and bridgeless, and remaining edges within contracted pieces can receive arbitrary directions.^[ref]