Cyclopian Edges

You are given a simple graph with $N$ nodes and $M$ edges. The graph may not be connected. Each edge in the graph has an integer weight assigned to it and each node is **coloured** either white or black. In one operation, you can choose an edge that has a white node at one end and a black node at the other, and reduce its weight by one. After this operation, the colour of both the nodes at the en