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Consider a graph 'g':

g = KaryTree[50, 3, VertexShapeFunction -> "Name"];
g1 = FindGraphPartition[g, 2]
g2 = HighlightGraph[g, Map[IGWeightedSubgraph[g, #] &, g1]]

enter image description here

Why doesn't the 'FindGraphPartition' command split the 'g' graph into two independent and coherent (connected) graphs? You can see that e.g. two red graphs are not connected. How to split a graph (especially weighted) into two approximately equal parts in the middle of the graph? (e.g. on the GraphCenter[g][[1]] node)

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