  We give the analysis of the computational complexity of coalition structure
generation over graphs. Given an undirected graph G = (N,E) and a valuation
function v : P(N) \to R over the subsets of nodes, the problem is to find a
partition of N into connected subsets, that maximises the sum of the components
values. This problem is generally NP-complete; in particular, it is hard for a
defined class of valuation functions which are independent of disconnected
members - that is, two nodes have no effect on each other's marginal
contribution to their vertex separator. Nonetheless, for all such functions we
provide bounds on the complexity of coalition structure generation over general
and minor-free graphs. Our proof is constructive and yields algorithms for
solving corresponding instances of the problem. Furthermore, we derive linear
time bounds for graphs of bounded treewidth. However, as we show, the problem
remains NP-complete for planar graphs, and hence, for any K_k minor free graphs
where k \geq 5. Moreover, a 3-SAT problem with m clauses can be represented by
a coalition structure generation problem over a planar graph with O(m^2) nodes.
Importantly, our hardness result holds for a particular subclass of valuation
functions, termed edge sum, where the value of each subset of nodes is simply
determined by the sum of given weights of the edges in the induced subgraph.

Data Structures and Algorithms;Computer Science and Game Theory