Consider the following problem:
Given an integer $k$ and a vertex-weighted graph $G=(V,E)$, find a partition of $V$ into $V_1,\ldots,V_k$ such that each subgraph induced by $V_i$ is connected, and such that $\max_i w(V_i)$ is minimized, where $w(V_i)$ is the total weight of the vertices in $V_i$.
It can be easily seen that this problem is NP-complete. For hardness, note that we can solve Partition by using a complete graph.
Questions: Does this problem have a standard name? Are there any papers that study this problem, or maybe slight variations of it? Is the problem still hard for sparse graphs? I am particularly interested in references that could lead to an implementation with good performance in practice.