Regular Cayley maps on dihedral groups with the smallest kernel

Let $\mathcal{M}=CM(D_n,X,p)$ be a regular Cayley map on the dihedral group $D_n$ of order $2n,n\ge 2,$ and let $\pi$ be the power function associated with $\mathcal{M}$. In this paper it is shown that the kernel Ker$(\pi )$ of the power function $\pi$ is a dihedral subgroup of $D_n$ and if $n\mathrm{\ne }3,$ then the kernel Ker$(\pi )$ is of order at least $4$. Moreover, all $\mathcal{M}$ are classified for which Ker$(\pi )$ is of order $4$. In particular, besides $4$ sporadic maps on $4,4,8$ and $12$ vertices respectively, two infinite families of non-$t$-balanced Cayley maps on $D_n$ are obtained.