The classification of two-distance transitive dihedrants

A vertex transitive graph Γ is said to be $2$-distance transitive if for each vertex u, the group of automorphisms of Γ fixing the vertex u acts transitively on the set of vertices at distance $1$ and $2$ from u, while Γ is said to be $2$-arc transitive if its automorphism group is transitive on the set of $2$-arcs. Then $2$-arc transitive graphs are $2$-distance transitive. The classification of $2$-arc transitive Cayley graphs on dihedral groups was given by Du, Malni{c} and Maru{s}i{c} in [Classification of 2-arc-transitive dihedrants, J. Combin. Theory Ser. B 98 (2008), 1349--1372]. In this paper, it is shown that a connected 2-distance transitive Cayley graph on the dihedral group of order $2n$ is either $2$-arc transitive, or isomorphic to the complete multipartite graph Km[b] for some m≥3 and b≥2 with $mb=2n$.