FIG. 4: Positions of the gluons corresponding to the two gravitons ( N + 1)$_{h}$ and ( N + 2)$_{h}$ for (a) 1 < l < i , 1 < k < i , (b) i < l ≤ N , i < k ≤ N in (c) i < l ≤ N , 1 < k < i and (d) 1 < l < i , i < k ≤ N in C part. Here we first insert the two gluons corresponding to ( N + 2)$_{h}$ between 1, l and l , i respectively. We then insert two gluons corresponding to ( N + 1)$_{h}$ between 1, k and k , i respectively. C. D$_{2}$ relations for arbitrary tree amplitudes with N gluons minimally coupled to M gravitons The amplitudes with more gravitons are more complicated. However the discussions are similar with those for amplitudes with one and two gravitons. The amplitude with N gluons minimally coupled to M gravitons, where two gluons take negative helicity and other legs 20