I've been looking for an algorithm which divides an undirected, weighted, planar and simple graph into $k$ disjoint subgraphs. Here, the graph is sparse, $k$ is fixed, and there are no negative edge weights. After cutting, each subgraph must be connected (i.e. there must be a path between any two vertices of the subgraph which is only composed of vertices in that subgraph).
However, unlike most existing work on graph partitioning out there, I don't intend to obtain subgraphs that contain the same approximate number of vertices. Instead, I would like these subgraphs to have similar sum of edge weights. In other words, I would like to minimize the sum of edge weights of the subgraph with maximal weight and ideally cut long (weighted) edges.
Is there a name for this problem? I wasn't able to find anything about this on the web. Also, how can I approach this problem?