I have a graph $G$ with $n$ nodes and $O(n)$ edges. Each of the nodes has a positive integral weight at least 2, such that sum of all weights is at most $O(n)$. We want to color the nodes and edges red or blue, where coloring a node of weight $C$ costs $C$ units, and coloring an edge costs 1 unit.
Constraints:
- Any edge between two blue nodes must be blue
- Any edge between two red nodes must be red
There are no constraints on edges between a red and a blue node.
Also given is we have exactly $R$ red and $B$ blue units of color, such that $R + B$ = sum of weights of nodes + edges. (Again, edges have weight 1.)
The problem is to determine whether the graph can be colored in a way that meets these constraints. Is this problem NP-complete? I suspect it is, but I am not sure which problem I can use to reduce this from.
