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.

Some Definitions:

  • 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.

2 Answers 2


I don't have the book handy. I assume that

  1. "safe" means that $(u,v)$ can be added to $A$ and we get another subset of a minimal spanning tree of $G$, and
  2. the goal is a correctness proof of Prim's algorithm¹.

Now, say the cut did not "respect" $A$. That means that there already is an edge in $A$ that crosses it. By choosing a second edge from that cut -- which may happen even with pairwise distinct weights! -- we may introduce a cycle and lose the tree property.

enter image description here

By requiring that the cut "respects" $A$ we know that $A \subseteq S$ and we can not introduce a cycle².

The assumptions of this theorem may be stronger than strictly necessary -- as long as we do not create a cycle, we do not per se need to care about other edges in the cut -- but we can build an algorithm around them, so the theorem is a convenient way to ensure its correctness.

  1. Note how the theorem directly implies that fact since Prim's algorithm maintains such a cut and chooses light edges from it to grow the tree.
  2. In the algorithm, we'll always have $A=S$.
  • $\begingroup$ Thanks. I agree with your opinion that cycle may be produced if the "respect" requirement is not satisfied. However, how could a cycle be produced when weights are pairwise distinct? If so, the heaviest edge in the cycle (which is not the edge$(u,v)$ because it is not the lightest edge across the cut) cannot be in any MST, and thus cannot be in $A$, contradicting the assumption that $A$ is a subset of $E$ that is included in some MST. $\endgroup$
    – hengxin
    May 24, 2016 at 12:28
  • $\begingroup$ The other edge of $u$ may have been chosen after that other one that already crosses the cut -- let's call it $e$ -- so we never compared $e$ with $(u,v)$. So the scenario can also happen with pairwise distinct weights. $\endgroup$
    – Raphael
    May 24, 2016 at 12:40
  • $\begingroup$ Do you mean that the lightest edge $e$ is chosen from the set of edges across the cut except those that are in $A$ but do not respect the cut? For example, suppose that $e' = (x,y) \in A$ with $w(e') = 2$ does not respect the cut $C$ and that $e'' = (x, z) \notin A$ with $w(e'') = 5$ is the only edge across the cut $C$ besides $e'$. Then the lightest edge across the cut at this stage is $e''$ instead of $e'$. In this case, $A \cup \{ e'' \}$ may contain a cycle (while $A \cup \{ e' \} = A$). $\endgroup$
    – hengxin
    May 24, 2016 at 13:01

Besides the effect of "avoiding cycles" pointed out by @Raphael, the "respect" requirement in the cut property guarantees that $A \cup \{ (u,v) \}$ is included in the minimum spanning tree $T' = T - \{ (x,y) \} \cup \{ (u,v) \}$ constructed in the correctness proof.


Since $A \subseteq T$, to ensure that $A \cup \{ (u,v) \} \subseteq T' = T - \{ (x,y) \} \cup \{ (u,v) \}$, it requires $\{ (x,y) \} \notin A$; see the figure above (Fig 23.3 in CLRS).

$(x,y) \notin A$ holds because $(x,y)$ is chosen as the edge that is on the unique path from $u$ to $v$ in $T$ and crosses the cut $(S, V-S)$ and that $A$ respects the cut $(S,V-S)$.


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