Variations of the maximum leaf spanning tree problem for bipartite graphs

The maximum leaf spanning tree problem is known to be NP-complete. In [M.S. Rahman, M. Kaykobad, Complexities of some interesting problems on spanning trees, Inform. Process. Lett. 94 (2005) 93-97], a variation on this problem was posed. This variation restricts the problem to bipartite graphs and asks, for a fixed integer K, whether or not the graph contains a spanning tree with at least K leaves in one of the partite sets. We show not only that this problem is NP-complete, but that it remains NP-complete for planar bipartite graphs of maximum degree 4. We also consider a generalization of a related decision problem, which is known to be polynomial-time solvable. We show the problem is still polynomial-time solvable when generalized to weighted graphs.