Optimal clustering of multipartite graphs

Given a graph G=(X,U)G=(X,U), the problem dealt within this paper consists in partitioning X   into a disjoint union of cliques by adding or removing a minimum number z(G)z(G) of edges (Zahn's problem). While the computation of z(G)z(G) is NP-hard in general, we show that its computation can be done in polynomial time when G is bipartite, by relating it to a maximum matching problem. When G   is a complete multipartite graph, we give an explicit formula specifying z(G)z(G) with respect to some structural features of G. In both cases, we give also the structure of all the optimal clusterings of G.