A graph $G$ contains a graph $H$ as an induced minor if $H$ can be obtained from $G$ by vertex deletions and edge contractions. The class of $H$-induced-minor-free graphs generalizes the class of $H$-minor-free graphs, but unlike $H$-minor-free graphs, it can contain dense graphs. We show that if an $n$-vertex $m$-edge graph $G$ does not contain a graph $H$ as an induced minor, then it has a balanced vertex separator of size $O_{H}(\sqrt{m})$, where the $O_{H}(\cdot)$-notation hides factors depending on $H$. More precisely, our upper bound for the size of the balanced separator is $O(\min(|V(H)|^2, \log n) \cdot \sqrt{|V(H)|+|E(H)|} \cdot \sqrt{m})$. We give an algorithm for finding either an induced minor model of $H$ in $G$ or such a separator in randomized polynomial-time. We apply this to obtain subexponential $2^{O_{H}(n^{2/3} \log n)}$ time algorithms on $H$-induced-minor-free graphs for a large class of problems including maximum independent set, minimum feedback vertex set, 3-coloring, and planarization. For graphs $H$ where every edge is incident to a vertex of degree at most 2, our results imply a $2^{O_{H}(n^{2/3} \log n)}$ time algorithm for testing if $G$ contains $H$ as an induced minor. Our second main result is that there exists a fixed tree $T$, so that there is no $2^{o(n/\log^3 n)}$ time algorithm for testing if a given $n$-vertex graph contains $T$ as an induced minor unless the Exponential Time Hypothesis (ETH) fails. Our reduction also gives NP-hardness, which solves an open problem asked by Fellows, Kratochv\'il, Middendorf, and Pfeiffer [Algorithmica, 1995], who asked if there exists a fixed planar graph $H$ so that testing for $H$ as an induced minor is NP-hard.

翻译：图$G$包含图$H$作为诱导子式，如果$H$可通过从$G$中删除顶点和收缩边得到。$H$-诱导子式排除图类推广了$H$-子式排除图类，但与$H$-子式排除图不同，前者可包含稠密图。我们证明：若$n$顶点$m$边图$G$不包含图$H$作为诱导子式，则存在大小为$O_{H}(\sqrt{m})$的平衡顶点分离器，其中$O_{H}(\cdot)$记号隐藏了依赖$H$的因子。更精确地，平衡分离器大小的上界为$O(\min(|V(H)|^2, \log n) \cdot \sqrt{|V(H)|+|E(H)|} \cdot \sqrt{m})$。我们给出一个随机多项式时间算法，可在$G$中寻找$H$的诱导子式模型或此类分离器。将此结果应用于$H$-诱导子式排除图，可对包括最大独立集、最小反馈顶点集、3-着色和平面化在内的广泛问题类获得次指数$2^{O_{H}(n^{2/3} \log n)}$时间算法。对于每条边至多与一个度数不超过2的顶点关联的图$H$，我们的结果蕴含一个$2^{O_{H}(n^{2/3} \log n)}$时间算法，用于判定$G$是否包含$H$作为诱导子式。第二个主要结果是：存在固定树$T$，使得除非指数时间假设（ETH）失效，否则不存在$2^{o(n/\log^3 n)}$时间算法判定给定$n$顶点图是否包含$T$作为诱导子式。我们的归约还给出NP难度，解决了Fellows、Kratochvíl、Middendorf和Pfeiffer [Algorithmica, 1995] 提出的公开问题——他们询问是否存在固定平面图$H$使得判定$H$作为诱导子式为NP难问题。