The classification of two-distance transitive dihedrants

A vertex transitive graph is said to be 2-distance transitive if for each vertex u, the group of automorphisms of fixing the vertex u acts transitively on the set of vertices at distance 1 and 2 from u, while 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, Malnic and Marusic 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 K₌₁ for some m3 and b2 with mb=2n.