Domination Critical Knodel Graphs

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 $1\le \Delta \le \lfloor \log_2n \rfloor$,aKn\ddot{o}delgraph$W_{\Delta,n}$isa$\Delta$-regularbipartitegraphofevenorder$n$,withvertices$(i,j)$,for$i=1,2$and$0\le j \le n/2-1$,whereforevery$j$,$0\le j \le n/2-1$, there is an edge between vertex $(1,j)$ and every vertex $(2,j+2^k-1$ (mod (n/2)), for $k=0,1,\cdots ,\mathrm{\Delta }-1$. 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.