Given an undirected connected graph, our goal is to remove some edges to make the graph disconnected. The constraint is that each node of the graph can not lose more than $m$ edges incident to it. I want to find the minimum $m$ for which the goal is achievable. Is there any efficient algorithm to compute this minimum $m$ (and/or which edges to remove)? Or is it NPcomplete?
It appears to be NPcomplete even when m=1: see The Complexity of the MatchingCut Problem, Maurizio Patrignani and Maurizio Pizzonia, WG 2001, http://dx.doi.org/10.1007/3540454772_26 


According to wikipedia, if your only constraint is minimizing the number of edges removes, it's easy. Now whether those algorithms are approriate for solving your problem as stated exactly or approximately, thats another question entirely. I'd guess that it's certainly easy (in at least one of an exact or approximate sense) as long as the graphs have some nice sparseness or geometric structure such as being planar or k regular. 

