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Joined 2026-09-21 Message
Incoming links: On Orientations and Shortest Paths by Refael Hassin and others
Robbins' theorem 2026-10-14
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]
On Orientations and Shortest Paths by Refael Hassin and others. Hassin and Megiddo, On Orientations and Shortest Paths (1989), printed pp. 591–592: bridge-connectedness, strong connectivity and Theorem 2.6 (Robbins)
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]
On Orientations and Shortest Paths by Refael Hassin and others. Hassin and Megiddo (1989), Theorem 2.6 proof, printed p. 592: cycle orientation and contraction, explicitly described as a different constructive proof from Robbins's