take positive helicity (15) and (16) can be given by a sum of terms. Each term has the form where each l$_{i}$ can be any positive helicity gluon, and n 1 $_{1}$, n 1 $_{2}$,..., n 2 $_{1}$, n 2 $_{2}$,..., ..., n N 1 , n N 2 ,... is a permutation of all the gravitons. Following the discussions on amplitudes with one and two gravitons, this term can be given by sum of MHV amplitudes with N + 2 M gluons with appropriate factors. The first line insert two gluons corresponding to n 1 between 1, l$_{1}$ and l$_{1}$ , i respectively, then insert two gluons corresponding to n 1 2 between 1, one gluon corresponding to n 1 and n 1 $_{1}$, the other gluon corresponding to n 1 respectlvely, ... After inserting all the gluons corresponding to gravitons in the first line, we insert two gluons corresponding to n 2 1 between 1, l$_{2}$ and l$_{2}$ , i respectively, then insert two gluons corresponding to n 2 between 1, one gluon corresponding to n 2 1 and n 2 $_{1}$, the other gluon corresponding to n 2 1 respectively, ... In this way, we insert all the 2 M gluons corresponding to the M gravitons into the amplitudes. The phase factor is s$_{l}$$_{1}$$_{,n}$ 1 $_{1}$s$_{n}$ 1 $_{1}$,n 1 $_{2}$...s$_{l}$$_{2}$$_{,n}$ 2 $_{1}$s$_{n}$ 2 $_{1}$,n 2 $_{2}$...s$_{l}$$_{N}$$_{,n}$ N 1 s$_{l}$$_{N}$$_{,n}$ N 2 ... . At last, the amplitudes become MHV tree amplitudes with N + 2 M gluons, where the two gluons corresponding to a same graviton take the same momentum. This is just the D$_{2}$ relations in field theory. In string theory, the D$_{2}$ relations are independent of helicity configurations of the legs. Then we expect the D$_{2}$ relations should have helicity-independent form. For example, in the relation for amplitudes with one graviton and N gluons given in Subsection IV A, we only sum over l correspongding to the gluons with positive helicity, and for each l , the two gluons 21 ig N − 2 ( − κ 2 ) M 〈 1 i 〉 4 〈 12 〉〈 23 ... 〈 N 1 〉 × 〈 l$_{1}$ 1 〉〈 l$_{1}$i 〉 [ l$_{1}$n 1 $_{1}$] 〈 n 1 $_{1}$1 n 1 $_{1}$i 〉〈 n 1 $_{1}$n 1 2 〉 × 〈 n 1 $_{1}$1 〉〈 n 1 $_{1}$i [ n 1 $_{1}$n 1 $_{2}$] n 1 $_{2}$1 n 1 $_{2}$i 〉〈 n 1 $_{1}$n 1 2 〉 × ... × 〈 l$_{2}$ 1 〉〈 l$_{2}$i 〉 [ l$_{2}$n 2 $_{1}$] 〈 n 2 $_{1}$1 n 2 $_{1}$i 〉〈 n 2 $_{1}$n 2 〉 × 〈 n 2 $_{1}$1 〉〈 n 2 $_{1}$i 〉 [ n 2 $_{1}$n 2 $_{2}$] n 2 $_{2}$1 n 2 $_{2}$i 〉〈 n 2 $_{1}$n 2 〉 × ... . × 〈 l$_{N}$ 1 〉〈 l$_{N}$i 〉 [ l$_{N}$n N 1 ] 〈 n N 1 1 n N 1 i 〉〈 n N 1 n N 2 〉 × 〈 n N 1 1 〉〈 n N 1 i 〉 [ n N 1 n N 2 ] n N 2 1 n N 2 i 〉〈 n N 1 n N 2 〉 × ..., (33)