  A set $D$ of vertices of a graph $G$ is a dominating set if each vertex of
$V(G)\setminus D$ is adjacent to some vertex of $D$. The domination number of
$G$, $\gamma(G)$, is the minimum cardinality of a dominating set of $G$. A
graph $G$ is called domination vertex critical, or just $\gamma$-critical if
removal of any vertex decreases the domination number. A graph $G$ is called
domination vertex stable, or just $\gamma$-stable, if removal of any vertex
does not decrease the domination number. For an even integer $n\ge 2$ and $1\le
\Delta \le \lfloor \log_2n \rfloor$, a Kn\"odel graph $W_{\Delta,n}$ is a
$\Delta$-regular bipartite graph of even order $n$, with vertices $(i,j)$, for
$i=1,2$ and $0\le j \le n/2-1$, where for every $j$, $0\le j \le n/2-1$, there
is an edge between vertex $(1,j)$ and every vertex $(2,j+2^k-1$ (mod (n/2)),
for $k=0,1,\cdots,\Delta-1$. in this paper, we study the domination criticality
and domination stability of Kn\"odel graphs. We charactrize the 3-regular and
4-regular Kn\"odel graphs by $\gamma$-criticality or $\gamma$-stability.
