Why is greedy algorithm not finding maximum independent set of a bipartite graph?

Question

I was trying to solve the maximum independent set problem on bipartite graphs using the greedy method. So came across this post which does exactly what i was trying to do. But am concentrating only on the bipartite graphs. The counter case in the answer is not a bipartite graph. Are there any bipartite graphs that this one wont work?

Greedy(G):
 S = {}
 While G is not empty:
 Let v be a node with minimum degree in G
 S = union(S, {v})
 remove v and its neighbors from G
return S

Why is greedy algorithm not finding maximum independent set of a graph?

Answer 1

The same approach as in the original question answer applies here as well, with a slightly modified graph:

Start by removing #5, What's left is a paw graph (nodes (1,3,4,7)). Remove its leaves in any order. You discover a four-node independent set: (1,3,5,7)

Start by removing #6. What's left is a 4-cycle. Removing any node forces either of these sets:

1. (1,3,6)
2. (2,4,6)

both are three-element maximal (as in, cannot be expanded) independent sets, and thus not maximum (as in, the largest possible).