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Let G be a simple graph of order n. The domination polynomial of G is the polynomial D(G,x)=∑_i=1ⁿd(G,i)xⁱ, where d(G,i) is the number of dominating sets of G of size i. A root of D(G,x) is called a domination root of G. We denote the set of distinct domination roots by Z(D(G,x)). Two graphs G and H are said to be D-equivalent, written as G∼H, if D(G,x)=D(H,x). The D-equivalence class of G is [G]={H:H∼G}. A graph G is said to be D-unique if [G]={G}. In this paper, we show that if a graph G has two distinct domination roots, then Z(D(G,x))={−2,0}. Also, if G is a graph with no pendant vertex and has three distinct domination roots, then Z(D(G,x))⊆{0,−2±2i,−3±3i2}. Also, we study the D-equivalence classes of some certain graphs. It is shown that if n≡0,2(mod3), then C_n is D-unique, and if n≡0(mod3), then [P_n] consists of exactly two graphs.