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with this node setup, I can select one edge per island, the problem is I don't know how to define or randomize the selected edge.

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enter image description here

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To select a random edge per mesh island, all you have to do is count the number of edges per mesh island, generate a random value between $0$ and $Edge Count -1$ and compare this with the edge index per mesh island.

Something like this:


(Blender 4.0.2+)

...or even more simplified:

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