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The min-rank of a digraph was shown to represent the length of an optimal scalar linear solution of the corresponding instance of the Index Coding with Side Information (ICSI) problem. In this paper, the graphs and digraphs of near-extreme min-ranks are studied. Those graphs and digraphs correspond to the ICSI instances having near-extreme transmission rates when using optimal scalar linear index codes. In particular, it is shown that the decision problem whether a digraph has min-rank two isdoi:10.1109/isit.2012.6283852 dblp:conf/isit/DauSC12 fatcat:viohjeqm6bhqfbthzflea5pjl4