On the complexity of graph clustering with bounded diameter

In this paper, we study the graph clustering (partition) problem. The problem is to determine whether there is a partition of the vertices of a graph into certain number of clusters such that the diameter of subgraph induced by each cluster does not exceed a prescribed bound. An amusing result shows that for chordal graphs (respectively, dually chordal graphs) the problem is NP-complete if the diameter bound is restricted to any even integer (respectively, to any odd integer); otherwise, the problem is polynomial solvable for both classes of graphs. Moreover, by a simple reduction using graph powers, we show that there is a unified approach for solving this problem in various graph classes, including distance-hereditary graphs, doubly chordal graphs, circular-arc graphs, and AT-free graphs.