Idea

Very simple idea:

• Initially, all vertices are on the far side of the cut.
• Pick one r to be on near side and check the distance of r to all its neighbors.
• From the set of vertices on the far side, pick the one u that is reachable from the near side over the smallest edge.
• Move u to the near side. Update the distances of u to its neighbors on the far side to reflect the edges in the cut.
• Repeat.

Example...