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The cut and cycle properties of the minimum spanning tree are well-known. It is easy to use similar arguments to prove them. But I wonder if one property be derived from the other.

Cut property: Suppose edges X are part of an MST of G=(V, E). Pick any subset of nodes S for which X does not cross between S and V-S, and let e be the lightest edge across this partition. Then X + {e} is part of some MST.

Cycle property: Pick any cycle in the graph, and let e be the heaviest edge in that cycle. Then, there is an MST that does not contain e.

From Dashupta, Papadimitrionu, and Vazirani's Algorithms

Tom Bennett
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  • I am not sure what you are asking. Since those properties are both true, yeah, from a logic point of view, they imply each other (since true implies true). – Nathaniel Feb 28 '24 at 22:43
  • I meant to ask if one property can be derived from the other. – Tom Bennett Feb 28 '24 at 23:06

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The technical answer is that they imply each other. Since they are both true statements either implication evaluates to $T \to T$ which is a true statement. However, I know that you are really asking if you can prove MSTs have the cycle property only using the fact that they have cut property. The answer is no.

We define a 1-tree to be a connected graph with exactly one cycle. Informally, they are trees plus an edge. Then minimum spanning 1-trees (MS1T) of a graph $G$ satisfy the cut property, but not necessarily the cycle property. Thus the cut property is not enough to prove the cycle property.

Also, the cycle property clearly is not enough to prove the cut property, since the minimum cost acyclic subgraphs (which will always be the graph with no edges whenever all of the edges have positive weights) satisfy the cycle property but not the cut property.

Essentially, the cut property arises from the spanning nature of trees (though not every class of spanning subgraph will satisfy this property) and the cycle property arises from the acyclic nature of trees (though not every class of acyclic subgraph will satisfy this property). Trees are special because they are spanning and acyclic.

NaturalLogZ
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