I think there is a mistake in the answer (from the book) of this problem.
The answer is:
{0},d=1
{1},d=0
{2,3,4,5}, d=1
I think state 1 must be with the states {2,3,4,5} and therefore with d=1.
Am I correct?
2026-02-23 20:36:08.1771878968
About the communicating classes
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No. Notice that, the states $\{2,3,4,5\}$ form a closed communicating class. State 1 is not communicating with any of the states in the class $\{2,3,4,5\}$. State 1 itself forms a class that is non-closed. It is non-closed because there is an escape from state 1 to state 2. When one the chain moves from state 1 to state 2, it will never visit state 1.