Except for the star and the single-edge graph, every other graph has inducibility at most c + o_k(1) for a fixed c < 1/e, and graphs with inducibility bounded away from zero are exactly those with a bounded-size taming set.
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Characterizing graphs with high inducibility
Except for the star and the single-edge graph, every other graph has inducibility at most c + o_k(1) for a fixed c < 1/e, and graphs with inducibility bounded away from zero are exactly those with a bounded-size taming set.