Vertex fusion under diameter constraints 1

Given a graph G=(V,E), a positive integer k, and a positive integer d, we want find a subset Vk with k vertices such the graph obtained by identifying the vertices from Vk in G has diameter at most d. We prove that for every d⩾2 the problem is NP-complete. For the case of trees we provide a polynomial time algorithm that exploits the relationship with the r-dominating set problem.