The cut property stated in terms of Theorem 23.1 in Section 23.1 of CLRS (2nd edition) is as follows.
Theorem 23.1 Let $G = (V, E)$ be a connected, undirected graph with a real-valued weight function $w$ defined on $E$. Let $A$ be a subset of $E$ that is included in some minimum spanning tree for $G$, let $(S, V-S)$ be any cut of $G$ that respect $A$ (emphasis added), and let $(u,v)$ be a light edge crossing $(S,V-S)$. Then, edge $(u,v)$ is safe for $A$.
Why does this theorem require that the cut $(S,V-S)$ respect $A$? How is this requirement used in the correctness proof? I do not see what would fail if the requirement was removed.
- Cut: A cut $(S, V-S)$ of an undirected graph $G=(V,E)$ is a partition of $V$.
- Cross: An edge $(u,v) \in E$ crosses the cut $(S,V-S)$ if one of its endpoints is in $S$ and the other is in $V-S$.
- Respect: A cut respects a set $A$ of edges if no edge in $A$ crosses the cut.
- Light edge: An edge is a light edge crossing a cut if its weight is the minimum of any edge crossing the cut.