Maximisation of graphs weight
It is definitely not a classic graph problem such as the shortest path, maximum-flow, or graph coloring problem. But your problem, in its general form, seems to solve the Independent set problem, which is $NP$-complete. 

Assume that you have a polynomial time algorithm $A(G,m,p)$ solving this problem. Then take $m=1$ and call $A(G,1,p)$ for $p=N,N-1,...,2,1$ (at most $N$ times) until you find a solution. When $m=1$, each subgraph has only one node and so its weight is 0, which is maximum for a single node graph. Furthermore, these nodes are not connected by an edge, and so they are not adjacent, i.e. they are independent. 

If this procedure finds a solution for the maximum $p$ less than $N$, then this means we have a maximal independent set, since 
we have exactly $n$ nodes (subgraphs) with no edges between them, i.e. independent.

Once your problem is $NP$-complete, you could try any approximation algorithm including Linear or Integer programming.