Why does the order in $(u,v)$ matter?

Conditions

• The graph is a directed graph.
• Edges are represented as ordered pairs.

Reasoning, step by step

1. Recall that a flow network is modeled as a directed graph.
2. Observe the definition of an edge $(u,v)$ as pointing from $u$ to $v$.
3. Note that direction is an inherent property of the edge in this model.
4. Conclude that swapping $u$ and $v$ changes the direction of the edge, thus referring to a different edge.

Example

The presenter draws a sketch with two vertices labeled $u$ and $v$ connected by an arrow from $u$ to $v$, explicitly stating that edge $uv$ means an edge pointing from node $u$ to node $v$ because the graph is directed.

Common misconceptions

• Treating $(u,v)$ as an unordered pair, as in undirected graphs.
• Assuming that an edge from $u$ to $v$ implies an automatic edge from $v$ to $u$ with the same capacity.

Watch the explanation

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