6.2.1. Circulations

A circulation in a network \(G\) is a pseudo-flow \(f\) that satisfies

\[\sum_{y \in V} \bigl(f_{x,y} - f_{y,x}\bigr) = 0 \quad \forall x \in V.\]

In other words, no vertex produces or absorbs any flow; the flow simply circulates through the network.

A cycle flow is a circulation \(f\) in \(G\) such that the edges \(e \in G\) with \(f_e > 0\) form a simple cycle in \(G\).

A path flow is a pseudo-flow \(f\) in \(G\) such that the edges \(e \in G\) with \(f_e > 0\) form a simple path \(P\) in \(G\) and, for every vertex \(x\) of \(G\) except the endpoints of \(P\),

\[\sum_{y \in V} \bigl(f_{x,y} - f_{y,x}\bigr) = 0.\]