I understand the basic meaning to communication classes however I am not 100% sure when the nodes match up slightly differently to the basic idea.
State Space $S = {1,2,3,4,5}$
$\left( \begin{array}{ccccc} 1/6 & 2/3 & 0 & 1/6 & 0 \\ 1/4 & 0 & 3/4 & 0 & 0 \\ 0 & 0 & 1/3 & 1/3 & 1/3 \\ 1/2 & 0 & 0 & 1/4 & 1/4 \\ 0 & 0 & 0 & 4/5 & 1/5 \end{array}\right)$
Currently I am thinking there are two classes:
$C_1 = {1,2,4,5}$
$C_2 = {3}$
I am not sure if $C_1$ is correct as I am not 100% sure on how to split the classes up when there are multiple node connections as starting off we can easily see that 1,2 communicate between each other directly, so does 1,4 then considering 4,5 communicate directly would this class be as I've stated ${1,2,4,5}$
Does this make this reducible since we have $C_2$ on it's own?
Thanks in advance!