looking for counterexample for my algorithm for maximum independent set in Bipartite Graph

Consider the following graph:

bipartite graph

None of the vertices has degree 1, so we go straight to step 5. Your algorithm would return 5 as the maximum independent set size, as both $L$ and $R$ have 5 vertices. However, the actual maximum independent set is ${1, 2, 3, 6, 7, 8}$, which has 6 vertices.