Definition: $\kappa(G)$ is defined as the minimum number of vertices whose removal from $G $ results in a disconnected graph.A graph $G$ is said to be $k$-connected if $\kappa(G) ≥ k$.

Context: Given a adjacency matrix A of a $r$ regular graph $G$ . $G$ is $k$ connected.

The matrix A can be divided into 4 sub matrices based on adjacency of vertex $x \in G$. $A_x$ is the adjacency matrix of the graph $(G-x)$, where $C$ is the symmetric matrix of the graph created by vertices of $(G-x)$ which are adjacent to $x$ and $D$ is the symmetric matrix of the graph created by vertices of $(G-x)$ which are not adjacent to $x$. $C,D$ are sub-graphs of graph $G$,$|V(C)|>|V(D)|$ where $|V(C)|,|V(D)|$ are total vertices number of graph $C,D$ respectively.

$$ A_x = \left( \begin{array}{ccc} C & E & 0 \\ E^{T} & D & 1\\ 0 & 1 & 0\\ \end{array} \right) $$ again, this process can be done recursively, where $A=C$. This recursive process can be done maximum $log_2(|V(G)|)$ times.
Question: In this recursive process, is it possible to have an $E$ matrix as a zero matrix, i.e. , is it possible to have disconnected sub-graphs $C,D$ under the given condition that $G$ is $k$ connected ? This question is connected to this [ post][1].

Request: Please, inform if anything is unclear/undefined . Also, inform , if the question is miss-tagged.Thanks. [1]: http://math.stackexchange.com/questions/1240637/counting-problem-of-combinations-of-symmetric-matrix