I have asked almost same question earlier. I have been told that my question was poorly written, so I am trying to write it more clearly in this post. Also, this time I would be a little different in my approach, I will not use matrix and recursive process, instead, this time I describe the problem for first two consecutive steps. Despite my effort, if it is not clear, please, let me know.
Problem Description: $G$ is a $r$ regular graph. $G$ is $k$ connected( not a complete or cycle graph). A vertex of $G$ is $x_1$. All vertices which are not adjacent to $x_1$ create a sub-graph $C_1$. All vertices adjacent to $x_1$ create a sub-graph $, D_1 $. These $C_1, D_1 $ are $s_1 , t_1 $ regular graphs respectively (given condition) . A vertex of $D_1$ is $x_2$.
Using same method, based on adjacency of $x_2$ , $D_1$ can be divided.
All vertices which are not adjacent to $x_2$ create a sub-graph $C_2$. All vertices adjacent to $x_2$ create a sub-graph $, D_2 $. These $C_2, D_2 $ are $s_2 , t_2 $ regular graphs respectively(given condition).
Question: According to above given conditions/situation/context, is it possible to have no edge between graph $C_2$ and $ D_2 $?
Motivation: I am studying graph isomorphism problem. In a certain kind of construction, above described situation arises and increases search cases. I am trying to find out when this (no edge between sub-graphs $C_y, D_y$ ) can happen. I hope, this post is an appropriate MO question.