Critical Graphs in Index Coding

In this paper, we define critical graphs as minimal graphs that support a given set of rates for the index coding problem and study them for both the one-shot and asymptotic setups. For the case of equal rates, we find the critical graph with minimum number of edges for both one-shot and asymptotic cases. For the general case of possibly distinct rates, we show that for one-shot and asymptotic linear index coding, as well as asymptotic nonlinear index coding, each critical graph is a union of disjoint strongly connected subgraphs. On the other hand, we identify a non-USCS critical graph for a one-shot nonlinear index coding problem. Next, we identify a few graph structures that are critical. In addition, we show that the capacity region of the index coding is additive for union of disjoint graphs.