The Linear Deterministic model has enabled a more intimate understanding of multi-terminal information theory problems, in particular, interference channels. Here we explore an underlying combinatorial structure of this model in the context of a 2-user linear deterministic interference channel (LDIC) and demonstrate how the sum-capacity of this channel is related to this structure. We extend this understanding to LDICs with $K \geq 2$ users in a symmetric parameter setting. Next we look at 2-user LDICs with output feedback and show how capacity improvements over the non-feedback case is again related to the combinatorial structure. We further use this to find sum-capacity of 2-user parallel LDICs and also demonstrate that certain separable parallel LDICs may become inseparable with feedback.