Vertex fusion under diameter constraints

Abstract

Given a graph $G$=($V$,$E$), a positive integer $k$ and a positive integer $d$, we want find a subset $V_k$ with $k$ vertices such the graph obtained by identifying the vertices from $V_k$ in $G$ has diameter at most $d$. We prove that for every $d geq 2$ 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.