# A graph connectivity problem (restated)

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 NP-complete?

-

## 2 Answers

It appears to be NP-complete even when m=1: see The Complexity of the Matching-Cut Problem, Maurizio Patrignani and Maurizio Pizzonia, WG 2001, http://dx.doi.org/10.1007/3-540-45477-2_26

-
It's even NP-complete with m = 1 and restricted to planar graphs, see portal.acm.org/citation.cfm?id=1598870. This also gives some classes of graphs for which there is a polynomial-time algorithm, though I suspect the constants can be huge (bounded tree-width?!). –  Harrison Brown Dec 1 '09 at 6:03

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.

-