The sum capacity of a class of discrete memoryless interference channels is determined. This class of channels is defined analogous to the Gaussian Z-interference channel with weak interference; as a result, the sum capacity is achieved by letting the transceiver pair subject to the interference communicates at a rate such that its message can be decoded at the unintended receiver using single user detection. Moreover, this class of discrete memoryless interference channels is equivalent in capacity region to certain discrete degraded interference channels. This allows the construction of a capacity outer-bound using the capacity region of associated degraded broadcast channels. The same technique is then used to determine the sum capacity of the discrete memoryless interference channel with mixed interference. The above results allow one to determine sum capacities or capacity regions of several new discrete memoryless interference channels.