Consider a graph 'g':
g = KaryTree[50, 3, VertexShapeFunction -> "Name"];
g1 = FindGraphPartition[g, 2]
g2 = HighlightGraph[g, Map[IGWeightedSubgraph[g, #] &, g1]]
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)
FindGraphPartitionsuses METIS. See the METIS FAQ Why am I getting non-contiguous partitions?. Perhaps you can petition Wolfram Research to expose the option that forces partitions to be contiguous. $\endgroup$