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The multiplexing gain (MUXG) of the K-user interference channel (IC) with partially connected interfering links is analyzed. The motivation for the partially connected IC comes from the fact that not all interferences are equally strong in practice. We introduce the idea of the graphical representation of interference alignment (IA) condition. We show that for a K-user partially connected MIMO IC with M antennas at each transmitter and receiver, the transmit beamforming matrices achieving the MUXG of KM/2 can be generated sequentially almost surely if the interference channel has a graphical representation which follows certain criteria. We also present a novel scheme, where we use the idea of partially connected IC to obtain improvements in conventional distributed interference alignment algorithm for fully connected IC. The improvements are obtained in terms of greater average sum-rate and faster convergence.