Hardness and tractability of detecting connected communities

We say that there is a community structure in a graph when the nodes of the graph can be partitioned into groups (communities) such that each group is internally more densely connected than with the rest of the graph. However, the challenge seems to specify what is to be dense, and what is relatively more connected (there seems to exist a similar situation to what is a cluster in unsupervised learning). Recently,
Olsen [13] provided a general definition that is significantly more generic that others. We make two observations regarding such definition. First, we show that finding a community structure with k connected equal size communities is NP-complete. Then, we show that this problem can be solved efficiently on trees. Finally, we observed that every tree has a 2-community structure. This result is based on a reduction from an extremely popular heuristic (the GirvanNeumann algorithm [12]) for detecting communities in large networks.