Then the amplitudes on D$_{2}$ can be given by color decomposed form where i a $_{o}$( i = 1 , ..., N ) denote open string legs with Chan-Paton degrees of freedoms and j$_{c}$ ( j = N + 1 , ..., N + M ) denote closed string legs. σ runs over the set of non cyclic permutations of the open strings. Then D$_{2}$ relations give the partial amplitudes A ( σ (1$_{o}$) , ..., σ ( N$_{o}$ ) , ( N + 1)$_{c}$ , ..., ( N + M )$_{c}$) for a given permutation of open strings by pure open string amplitudes: where P ′′ are all the permutations of the N + 2 M external legs which preserve the relative positions of the open strings 1$_{o}$ , ..., N$_{o}$ . This expression implies that for a given permutation of the open strings on the boundary of D$_{2}$ , any closed string can split into two open strings inserted on the boundary of D$_{2}$ . Using contour deformations, the relations (5) can also be reduced[6]. In the reduced form of the relations, the phase factors become sine functions. In field theory limits, D$_{2}$ relations give tree amplitudes with N gluons and M gravitons by pure-gluon amplitudes with N + 2 M legs. They are different from the KLT factorization relations which factorize amplitudes with N + M legs into products of two amplitudes with N + M legs. For M = 0, D$_{2}$ relations trivially give the pure-gluon amplitudes. Then we do not need to consider M = 0 case. Since the generators of the gauge group satisfy T r ( T $^{a}$) = 0, we also do not need to consider M = 1 case. After taking the field theory limit α $^{′}$→ 0, the D$_{2}$ relations for two-gluon one-graviton tree amplitude A (1$_{g}$ , 2$_{g}$ , 3$_{h}$), three-gluon one-graviton tree amplitude A (1$_{g}$ , 2$_{g}$ , 3$_{g}$ , 4$_{h}$) and two-gluon two-graviton tree amplitude A (1$_{g}$ , 2$_{g}$ , 3$_{h}$ , 4$_{h}$) 6 M (1 a$_{1}$ o , ..., N a$_{N}$ o , ( N + 1)$_{c}$ , ..., ( N + M )$_{c}$) = ∑ σ T r ( T a$_{σ}$ $^{(1)}$...T a$_{σ}$ ( N $^{)}$) A ( σ (1$_{o}$) , ..., σ ( N$_{o}$ ) , ( N + 1)$_{c}$ , ..., ( N + M )$_{c}$) , (4) A ( σ (1$_{o}$) , ..., σ ( N$_{o}$ ) , ( N + 1)$_{c}$ , ..., ( N + M )$_{c}$) = ∑ P e iπ Θ $^{′}$( P $^{$^{′′}$)}$A ( N, 2 M $^{)}$( P $^{′′}$) . (5)