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On wikipedia it says that boruvka can be implemented in linear time for planar graphs, but I don't know how to prove that.

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    $\begingroup$ Wikipedia also seems to give a big hint for how one could do that. Did you try removing all but the cheapest edge between each pair of components after each stage of the algorithm? $\endgroup$
    – Juho
    Dec 10 '20 at 8:35

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