I'm learning the requirements for finding a minimum spanning tree for a connected, undirected graph with distinct edge costs. One of the requirements is that there must be no cycles created in the tree, and the reason given for why a cycle isn't created by, for example Prim's algorithm, is that "an edge which is the only edge crossing a cut cannot create a cycle" (Lonely cut corollary). However, when I look at a cut, I normally see multiple edges crossing the cut. They do not necessarily connect the same two vertices, but there are multiple edges nevertheless. Shouldn't the lonely cut corollary be worded as "an edge which is the only one crossing a cut and connecting two specific vertices in each set"? Or am I just misunderstanding the corollary?
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How is a cut lonely if there are often multiple edges crossing a cut in a connected undirected graph?
It isn't.
And thus the corollary doesn't say anything about it. It only says something about an edge IF you have a lonely cut.
Kelly Bundy
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Well then that's not a very helpful criteria for Prim's algorithm to avoid cycles when creating an MST. How are they to be avoided when there are multiple edges crossing a cut? – julieb Jan 18 '22 at 17:02
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I don't know your book or whatever you got this from, so I can't tell. And I don't see why one would use it for Prim's algorithm, which rather obviously doesn't create cycles. But apparently the book/whatever talks about something where you do have lonely cuts and where the corollary is useful. Anyway, that's an *application* of the corollary, and the corollary itself is fine as it is and doesn't need your rewording. – Kelly Bundy Jan 18 '22 at 22:26
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After going back and reviewing the material again, I realized I was conflating edges of the original graph with edges of the graph being selected to create the tree by Prim's algorithm. As long as an edge crossing a cut is the first one added by the algorithm to cross that particular cut, no cycles are being created and Prim outputs a tree.
julieb
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