8.997 Topics in Combinatorial Optimization 18 Orientations, Directed Cuts and Submodular Flows 18.1 Graph Orientations

Proof: ⇐: Fix a strongly-connected orientation D. For any non-empty U ⊂ V , we may choose u ∈ U and v ∈ V \U . Since D is strongly connected, there is a directed u-v path and a directed v-u δ in path. Thus |δ out (U)| ≥ 1 and | D (U)| ≥ 1, implying |δG(U)| ≥ 2. D ⇒: Since G is 2-edge-connected, it has an ear decomposition. We proceed by induction on the number of ears. If G is a cycle then we may orient the edges to form a directed cycle D, which is obviously strongly connected. Otherwise, G consists of an ear P and subgraph G′ with a strongly connected orientation D′ . The ear is an undirected path with endpoints x, y ∈ V (G′) (possibly x = y). We orient P so that it is a directed path from x to y and add this to D′, thereby obtaining an orientation D of G. To show that D is strongly connected, consider any u, v ∈ V (G). If u, v ∈ V (G′) then by induction there is a u-v dipath. If u ∈ P and v ∈ V (G′) then there is a u-y dipath and by induction there is a y-v dipath. Concatenating these gives a u-v dipath. The case u ∈ V (G′) and v ∈ P is symmetric. If both u, v ∈ P then either a subpath of P is a u-v path, or there exist a u-y path, a y-x path, and a x-v path. (The y-x path exists by induction). Concatenating these three paths gives a u-v path. �