The maximum number of edge-disjoint $s$-$t$ paths equals the minimum number of edges in an $s$-$t$ edge cut (minimum set of edges whose removal disconnects $s$ from $t$). This follows from max-flow min-cut with unit capacities. The vertex version: the maximum number of internally vertex-disjoint $s$-$t$ paths equals the minimum vertex cut size (for non-adjacent $s, t$). This is proved by "splitting" each vertex $v$ into $v_{\text{in}}, v_{\text{out}}$ with a unit-capacity edge between them.

In minimum cost flow, each edge has capacity $c(e)$ and cost $a(e)$. The goal: send $d$ units of flow from $s$ to $t$ minimizing total cost $\sum_e a(e)f(e)$. The successive shortest paths algorithm finds minimum cost augmenting paths (using Dijkstra or Bellman-Ford in the residual graph with reduced costs). The cycle-canceling algorithm starts with any feasible flow and eliminates negative-cost cycles. Both run in polynomial time.

Any feasible flow $f$ in a network can be decomposed into at most $|E|$ path flows (from $s$ to $t$) and cycle flows, each with positive value. The decomposition is not unique but always exists. This is useful for interpreting the combinatorial structure of an optimal flow: each path in the decomposition represents a "route" for transporting goods from source to sink.