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Let G be the dihedral group Dn and Cay(G, S) is the Cayley graph of G with respect to S, and let CG(x) is the centralizer of an element x in G and x is the orbit of x in G. In this paper, we prove that if G act on G by conjugation, the vertex induced subgraph with vertex set CG(x) of the Cayley graph Cay(G, x) is either Hamiltonian or it contain Hamiltonian decompositions. But if n is prime, it is always Hamiltonian. [ABSTRACT FROM AUTHOR]