We study the problem of finding the smallest graph that does not occur as an induced subgraph of a given graph. This missing induced subgraph has at most logarithmic size and can be found by a brute-force search, in an $n$-vertex graph, in time $n^{O(\log n)}$. We show that under the Exponential Time Hypothesis this quasipolynomial time bound is optimal. We also consider variations of the problem in which either the missing subgraph or the given graph comes from a restricted graph family; for instance, we prove that the smallest missing planar induced subgraph of a given planar graph can be found in polynomial time.

翻译：我们研究寻找给定图中未作为诱导子图出现的最小图的问题。该缺失诱导子图的大小最多为对数级别，并可通过暴力搜索在$n$个顶点的图中以$n^{O(\log n)}$时间找到。我们证明，在指数时间假设下，这一拟多项式时间界是最优的。我们还考虑了问题的变体，其中缺失子图或给定图来自受限图族；例如，我们证明给定平面图的最小缺失平面诱导子图可以在多项式时间内找到。