Original arrows show flow / capacity. Tables preserve every edge identity and number.
The residual path
Original edge assignments
| ID | Original edge | Flow / capacity | Add forward | Can cancel | This step |
|---|
See the available residual options
A use option adds flow to its original edge. A cancellation option removes flow from its own original edge. Options pointing the same way may belong to different originals.
| Residual direction | Changes this original edge | Available |
|---|
No positive residual options remain.
Check flow conservation
| Vertex | Incoming | Outgoing | Net out |
|---|
Intermediate vertices have net outflow zero. The source’s net outflow equals the sink’s net inflow.
The JSON includes the applied network, every trace step, the final cut and your selected inspection step. It is an algorithm observation, separate from a learner-answer archive.
What the proof says
A feasible assignment gives a value you can achieve. A cut gives a value you cannot exceed. When they meet, the maximum is known.
This explorer uses Edmonds–Karp: each augmentation follows a path with the fewest residual edges. It visits neighbors in the entered vertex order, preferring forward use before cancellation to the same next vertex, then original edge order. It does not choose by largest bottleneck or smallest cost.
All capacities and flows here are exact small integers. The network is a teaching model; its units and behavior are not measurements of pipes, traffic or another physical system.